Kato-Milne Cohomology and Polynomial Forms

Adam Chapman; Kelly McKinnie

arXiv:1705.09553·math.RA·March 2, 2018

Kato-Milne Cohomology and Polynomial Forms

Adam Chapman, Kelly McKinnie

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TL;DR

This paper explores the relationship between Kato-Milne cohomology, differential forms, and polynomial forms over fields of prime characteristic, establishing bounds on symbol length and conditions for cohomology triviality.

Contribution

It introduces the concept of $ ilde{C}_{p,m}$ fields and links $p$-regular forms to cohomological properties, providing new bounds and vanishing results.

Findings

01

Bound on symbol length of $H_p^2(F)$ by $p^{m-1}-1$

02

Vanishing of $H_p^{n+1}(F)$ for large $n$

03

Connection between $p$-regular forms and cohomology structure

Abstract

Given a prime number $p$ , a field $F$ with $char (F) = p$ and a positive integer $n$ , we study the class-preserving modifications of Kato-Milne classes of decomposable differential forms. These modifications demonstrate a natural connection between differential forms and $p$ -regular forms. A $p$ -regular form is defined to be a homogeneous polynomial form of degree $p$ for which there is no nonzero point where all the order $p - 1$ partial derivatives vanish simultaneously. We define a $C_{p, m}$ field to be a field over which every $p$ -regular form of dimension greater than $p^{m}$ is isotropic. The main results are that for a $C_{p, m}$ field $F$ , the symbol length of $H_{p}^{2} (F)$ is bounded from above by $p^{m - 1} - 1$ and for any $n \geq ⌈(m - 1) lo g_{2} (p)⌉ + 1$ , $H_{p}^{n + 1} (F) = 0$ .

Equations84

(a_{0} d a_{1} \land \dots \land d a_{n}) (b_{0} d b_{1} \land \dots \land d b_{m}) = a_{0} b_{0} d a_{1} \land \dots \land d a_{n} \land d b_{1} \land \dots \land d b_{m} .

(a_{0} d a_{1} \land \dots \land d a_{n}) (b_{0} d b_{1} \land \dots \land d b_{m}) = a_{0} b_{0} d a_{1} \land \dots \land d a_{n} \land d b_{1} \land \dots \land d b_{m} .

α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{n}}{β _{n}} ⟼ (α^{p} - α) \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{n}}{β _{n}} .

α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{n}}{β _{n}} ⟼ (α^{p} - α) \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{n}}{β _{n}} .

H_{2}^{n + 1} (F)

H_{2}^{n + 1} (F)

α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{n}}{β _{n}}

H_{p}^{2} (F)

H_{p}^{2} (F)

α \frac{d β}{β}

F ⟨ x, y : x^{p} - x = α, y^{p} = β, y x y^{- 1} = x + 1 ⟩ .

F ⟨ x, y : x^{p} - x = α, y^{p} = β, y x y^{- 1} = x + 1 ⟩ .

ω_{i} = α_{i} \frac{d β _{i, 1}}{β _{i, 1}} \land \dots \land \frac{d β _{i, n}}{β _{i, n}}, i \in {1, \dots, m}

ω_{i} = α_{i} \frac{d β _{i, 1}}{β _{i, 1}} \land \dots \land \frac{d β _{i, n}}{β _{i, n}}, i \in {1, \dots, m}

φ (a_{1} v_{1} + \dots + a_{m} v_{m}) = i_{1} + \dots + i_{m} = p \sum c_{i_{1}, \dots, i_{m}} a_{1}^{i_{1}} \dots a_{m}^{i_{m}}

φ (a_{1} v_{1} + \dots + a_{m} v_{m}) = i_{1} + \dots + i_{m} = p \sum c_{i_{1}, \dots, i_{m}} a_{1}^{i_{1}} \dots a_{m}^{i_{m}}

N_{\overline{F}} (a_{1} e_{1} + a_{2} e_{2} + \dots + a_{p} e_{p}) = a_{1} a_{2} \dots a_{p} .

N_{\overline{F}} (a_{1} e_{1} + a_{2} e_{2} + \dots + a_{p} e_{p}) = a_{1} a_{2} \dots a_{p} .

φ (a_{1} v_{1} + a_{2} v_{2}) = α a_{1}^{p} - a_{1} a_{2}^{p - 1} + a_{2}^{p}

φ (a_{1} v_{1} + a_{2} v_{2}) = α a_{1}^{p} - a_{1} a_{2}^{p - 1} + a_{2}^{p}

D = [α_{1}, β_{1})_{p, K} \otimes \dots \otimes [α_{n}, β_{n})_{p, K}

D = [α_{1}, β_{1})_{p, K} \otimes \dots \otimes [α_{n}, β_{n})_{p, K}

ω = α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{i}}{β _{i}} \land \dots \land \frac{d ( β _{i} β _{j} )}{β _{i} β _{j}} \land \dots \land \frac{d β _{n}}{β _{n}} .

ω = α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{i}}{β _{i}} \land \dots \land \frac{d ( β _{i} β _{j} )}{β _{i} β _{j}} \land \dots \land \frac{d β _{n}}{β _{n}} .

ω = α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d ( β _{i} + β _{j} )}{β _{i} + β _{j}} \land \dots \land \frac{d ( β _{i}^{- 1} β _{j} )}{β _{i}^{- 1} β _{j}} \land \dots \land \frac{d β _{n}}{β _{n}} .

ω = α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d ( β _{i} + β _{j} )}{β _{i} + β _{j}} \land \dots \land \frac{d ( β _{i}^{- 1} β _{j} )}{β _{i}^{- 1} β _{j}} \land \dots \land \frac{d β _{n}}{β _{n}} .

ω = α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{i}}{β _{i}} \land \dots \land \frac{d (( β _{i} + N _{F [℘^{- 1} (α)] / F} ( f )) β _{j} )}{( β _{i} + N _{F [℘^{- 1} (α)] / F} ( f )) β _{j}} \land \dots \land \frac{d β _{n}}{β _{n}} .

ω = α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{i}}{β _{i}} \land \dots \land \frac{d (( β _{i} + N _{F [℘^{- 1} (α)] / F} ( f )) β _{j} )}{( β _{i} + N _{F [℘^{- 1} (α)] / F} ( f )) β _{j}} \land \dots \land \frac{d β _{n}}{β _{n}} .

\frac{d β _{i}}{β _{i}} \land \frac{d β _{j}}{β _{j}} = \frac{d ( β _{i} + β _{j} )}{β _{i} + β _{j}} \land \frac{d ( β _{i}^{- 1} β _{j} )}{β _{i}^{- 1} β _{j}} .

\frac{d β _{i}}{β _{i}} \land \frac{d β _{j}}{β _{j}} = \frac{d ( β _{i} + β _{j} )}{β _{i} + β _{j}} \land \frac{d ( β _{i}^{- 1} β _{j} )}{β _{i}^{- 1} β _{j}} .

α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{i}}{β _{i}} \land \dots \land \frac{d (( β _{i} + N _{F [℘^{- 1} (α)] / F} ( f )) β _{j} )}{( β _{i} + N _{F [℘^{- 1} (α)] / F} ( f )) β _{j}} \land \dots \land \frac{d β _{n}}{β _{n}} .

α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{i}}{β _{i}} \land \dots \land \frac{d (( β _{i} + N _{F [℘^{- 1} (α)] / F} ( f )) β _{j} )}{( β _{i} + N _{F [℘^{- 1} (α)] / F} ( f )) β _{j}} \land \dots \land \frac{d β _{n}}{β _{n}} .

α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{i}}{β _{i}} \land \dots \land \frac{d ( β _{i} + N _{F [℘^{- 1} (α)] / F} ( f ))}{β _{i} + N _{F [℘^{- 1} (α)] / F} ( f )} \land \dots \land \frac{d β _{n}}{β _{n}} .

α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{i}}{β _{i}} \land \dots \land \frac{d ( β _{i} + N _{F [℘^{- 1} (α)] / F} ( f ))}{β _{i} + N _{F [℘^{- 1} (α)] / F} ( f )} \land \dots \land \frac{d β _{n}}{β _{n}} .

α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d N _{F [℘^{- 1} (α)] / F} ( f ) ^{- 1} β _{i}}{N _{F [℘^{- 1} (α)] / F} ( f ) ^{- 1} β _{i}} \land \dots \land \frac{d ( N _{F [℘^{- 1} (α)] / F} ( f ) ^{- 1} β _{i} + 1 )}{N _{F [℘^{- 1} (α)] / F} ( f ) ^{- 1} β _{i} + 1} \land \dots \land \frac{d β _{n}}{β _{n}},

α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d N _{F [℘^{- 1} (α)] / F} ( f ) ^{- 1} β _{i}}{N _{F [℘^{- 1} (α)] / F} ( f ) ^{- 1} β _{i}} \land \dots \land \frac{d ( N _{F [℘^{- 1} (α)] / F} ( f ) ^{- 1} β _{i} + 1 )}{N _{F [℘^{- 1} (α)] / F} ( f ) ^{- 1} β _{i} + 1} \land \dots \land \frac{d β _{n}}{β _{n}},

(\varphi,V)=\mathop{\mathchoice{\vbox{ \hbox{\scalebox{2.1}{$\displaystyle\perp$}} }}{\vbox{ \hbox{\scalebox{1.5}{$\textstyle\perp$}} }}{\vbox{ \hbox{\scalebox{1.5}{$\scriptstyle\perp$}} }}{\vbox{ \hbox{\scalebox{1.5}{$\scriptscriptstyle\perp$}} }}}\displaylimits_{\begin{array}[]{r}0\leqslant d_{1},\dots,d_{n}\leqslant 1\\ (d_{1},\dots,d_{n})\neq(0,\dots,0)\end{array}}(\varphi_{d_{1},\dots,d_{n}},V_{d_{1},\dots,d_{n}}).

(\varphi,V)=\mathop{\mathchoice{\vbox{ \hbox{\scalebox{2.1}{$\displaystyle\perp$}} }}{\vbox{ \hbox{\scalebox{1.5}{$\textstyle\perp$}} }}{\vbox{ \hbox{\scalebox{1.5}{$\scriptstyle\perp$}} }}{\vbox{ \hbox{\scalebox{1.5}{$\scriptscriptstyle\perp$}} }}}\displaylimits_{\begin{array}[]{r}0\leqslant d_{1},\dots,d_{n}\leqslant 1\\ (d_{1},\dots,d_{n})\neq(0,\dots,0)\end{array}}(\varphi_{d_{1},\dots,d_{n}},V_{d_{1},\dots,d_{n}}).

ω = α \frac{d β _{1}^{'}}{β _{1}^{'}} \land \dots \land \frac{d β _{n - 1}^{'}}{β _{n - 1}^{'}} \land \frac{d φ ( v )}{φ ( v )} .

ω = α \frac{d β _{1}^{'}}{β _{1}^{'}} \land \dots \land \frac{d β _{n - 1}^{'}}{β _{n - 1}^{'}} \land \frac{d φ ( v )}{φ ( v )} .

α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{n - 2}}{β _{n - 2}} \land \frac{d β _{n - 1}}{β _{n - 1}} = α \frac{d β _{1}^{'}}{β _{1}^{'}} \land \dots \land \frac{d β _{n - 2}^{'}}{β _{n - 2}^{'}} \land \frac{d φ ( τ ( v _{1}^{'} ))}{φ ( τ ( v _{1}^{'} ))}, and so

α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{n - 2}}{β _{n - 2}} \land \frac{d β _{n - 1}}{β _{n - 1}} = α \frac{d β _{1}^{'}}{β _{1}^{'}} \land \dots \land \frac{d β _{n - 2}^{'}}{β _{n - 2}^{'}} \land \frac{d φ ( τ ( v _{1}^{'} ))}{φ ( τ ( v _{1}^{'} ))}, and so

α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{n - 2}}{β _{n - 2}} \land \frac{d β _{n - 1}}{β _{n - 1}} \land \frac{d β _{n}}{β _{n}} = α \frac{d β _{1}^{'}}{β _{1}^{'}} \land \dots \land \frac{d β _{n - 2}^{'}}{β _{n - 2}^{'}} \land \frac{d φ ( τ ( v _{1}^{'} ))}{φ ( τ ( v _{1}^{'} ))} \land \frac{d β _{n}}{β _{n}} .

α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{n - 2}}{β _{n - 2}} \land \frac{d β _{n - 1}}{β _{n - 1}} \land \frac{d β _{n}}{β _{n}} = α \frac{d β _{1}^{'}}{β _{1}^{'}} \land \dots \land \frac{d β _{n - 2}^{'}}{β _{n - 2}^{'}} \land \frac{d φ ( τ ( v _{1}^{'} ))}{φ ( τ ( v _{1}^{'} ))} \land \frac{d β _{n}}{β _{n}} .

ω = α \frac{d β _{1}^{'}}{β _{1}^{'}} \land \dots \land \frac{d β _{n - 2}^{'}}{β _{n - 2}^{'}} \land \frac{d φ ( τ ( v _{1}^{'} ))}{φ ( τ ( v _{1}^{'} ))} \land \frac{d (( φ ( τ ( v _{1}^{'} )) + N _{F [℘^{- 1} (α)] / F} ( v _{0, 1} )) β _{n} )}{( φ ( τ ( v _{1}^{'} )) + N _{F [℘^{- 1} (α)] / F} ( v _{0, 1} )) β _{n}} .

ω = α \frac{d β _{1}^{'}}{β _{1}^{'}} \land \dots \land \frac{d β _{n - 2}^{'}}{β _{n - 2}^{'}} \land \frac{d φ ( τ ( v _{1}^{'} ))}{φ ( τ ( v _{1}^{'} ))} \land \frac{d (( φ ( τ ( v _{1}^{'} )) + N _{F [℘^{- 1} (α)] / F} ( v _{0, 1} )) β _{n} )}{( φ ( τ ( v _{1}^{'} )) + N _{F [℘^{- 1} (α)] / F} ( v _{0, 1} )) β _{n}} .

α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{n - 2}}{β _{n - 2}} \land \frac{d β _{n - 1}}{β _{n - 1}} \land \frac{d β _{n}}{β _{n}} = α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{n - 2}}{β _{n - 2}} \land \frac{d β _{n - 1}}{β _{n - 1}} \land \frac{d ( φ ( v _{1} ))}{φ ( v _{1} )} .

α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{n - 2}}{β _{n - 2}} \land \frac{d β _{n - 1}}{β _{n - 1}} \land \frac{d β _{n}}{β _{n}} = α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{n - 2}}{β _{n - 2}} \land \frac{d β _{n - 1}}{β _{n - 1}} \land \frac{d ( φ ( v _{1} ))}{φ ( v _{1} )} .

α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{n - 2}}{β _{n - 2}} \land \frac{d β _{n - 1}}{β _{n - 1}} = α \frac{d β _{1}^{''}}{β _{1}^{''}} \land \dots \land \frac{d β _{n - 2}^{''}}{β _{n - 2}^{''}} \land \frac{d φ ( v _{0} )}{φ ( v _{0} )}, and so

α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{n - 2}}{β _{n - 2}} \land \frac{d β _{n - 1}}{β _{n - 1}} = α \frac{d β _{1}^{''}}{β _{1}^{''}} \land \dots \land \frac{d β _{n - 2}^{''}}{β _{n - 2}^{''}} \land \frac{d φ ( v _{0} )}{φ ( v _{0} )}, and so

α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{n - 2}}{β _{n - 2}} \land \frac{d β _{n - 1}}{β _{n - 1}} \land \frac{d φ ( v _{1} )}{φ ( v _{1} )} = α \frac{d β _{1}^{''}}{β _{1}^{''}} \land \dots \land \frac{d β _{n - 2}^{''}}{β _{n - 2}^{''}} \land \frac{d φ ( v _{0} )}{φ ( v _{0} )} \land \frac{d φ ( v _{1} )}{φ ( v _{1} )} .

α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{n - 2}}{β _{n - 2}} \land \frac{d β _{n - 1}}{β _{n - 1}} \land \frac{d φ ( v _{1} )}{φ ( v _{1} )} = α \frac{d β _{1}^{''}}{β _{1}^{''}} \land \dots \land \frac{d β _{n - 2}^{''}}{β _{n - 2}^{''}} \land \frac{d φ ( v _{0} )}{φ ( v _{0} )} \land \frac{d φ ( v _{1} )}{φ ( v _{1} )} .

ω = α \frac{d β _{1}^{''}}{β _{1}^{''}} \land \dots \land \frac{d β _{n - 2}^{''}}{β _{n - 2}^{''}} \land \frac{d ( φ ( v _{0} ) ^{- 1} φ ( v _{1} ))}{φ ( v _{0} ) ^{- 1} φ ( v _{1} )} \land \frac{d ( φ ( v _{0} ) + φ ( v _{1} ))}{φ ( v _{0} ) + φ ( v _{1} )}

ω = α \frac{d β _{1}^{''}}{β _{1}^{''}} \land \dots \land \frac{d β _{n - 2}^{''}}{β _{n - 2}^{''}} \land \frac{d ( φ ( v _{0} ) ^{- 1} φ ( v _{1} ))}{φ ( v _{0} ) ^{- 1} φ ( v _{1} )} \land \frac{d ( φ ( v _{0} ) + φ ( v _{1} ))}{φ ( v _{0} ) + φ ( v _{1} )}

V_{1} = 0 ⩽ d_{1}, \dots, d_{n - 1} ⩽ 1 ⨁ V_{d_{1}, \dots, d_{n - 1}, 1} .

V_{1} = 0 ⩽ d_{1}, \dots, d_{n - 1} ⩽ 1 ⨁ V_{d_{1}, \dots, d_{n - 1}, 1} .

ω = α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{n - 1}}{β _{n - 1}} \land \frac{d φ ( v )}{φ ( v )} .

ω = α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{n - 1}}{β _{n - 1}} \land \frac{d φ ( v )}{φ ( v )} .

α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{n - 2}}{β _{n - 2}} \land \frac{d β _{n - 1}}{β _{n - 1}} = α \frac{d β _{1}^{'}}{β _{1}^{'}} \land \dots \land \frac{d β _{n - 2}^{'}}{β _{n - 2}^{'}} \land \frac{d φ ( τ ( v _{1}^{'} ))}{φ ( τ ( v _{1}^{'} ))},

α \frac{d β _{1}}{β _{1}} \land \dots \land \frac{d β _{n - 2}}{β _{n - 2}} \land \frac{d β _{n - 1}}{β _{n - 1}} = α \frac{d β _{1}^{'}}{β _{1}^{'}} \land \dots \land \frac{d β _{n - 2}^{'}}{β _{n - 2}^{'}} \land \frac{d φ ( τ ( v _{1}^{'} ))}{φ ( τ ( v _{1}^{'} ))},

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Full text

Kato-Milne Cohomology and Polynomial Forms

Adam Chapman

[email protected]

Department of Computer Science, Tel-Hai Academic College, Upper Galilee, 12208 Israel

Kelly McKinnie

[email protected]

Department of Mathematics, University of Montana, Missoula, MT 59812, USA

Abstract

Given a prime number $p$ , a field $F$ with $\operatorname{char}(F)=p$ and a positive integer $n$ , we study the class-preserving modifications of Kato-Milne classes of decomposable differential forms. These modifications demonstrate a natural connection between differential forms and $p$ -regular forms. A $p$ -regular form is defined to be a homogeneous polynomial form of degree $p$ for which there is no nonzero point where all the order $p-1$ partial derivatives vanish simultaneously. We define a $\widetilde{C}_{p,m}$ field to be a field over which every $p$ -regular form of dimension greater than $p^{m}$ is isotropic. The main results are that for a $\widetilde{C}_{p,m}$ field $F$ , the symbol length of $H_{p}^{2}(F)$ is bounded from above by $p^{m-1}-1$ and for any $n\geqslant\lceil(m-1)\log_{2}(p)\rceil+1$ , $H_{p}^{n+1}(F)=0$ .

keywords:

Decomposable Differential Forms, Quadratic Pfister Forms, Cyclic $p$ -Algebras, Quaternion Algebras, Kato-Milne Cohomology, Linkage

MSC:

[2010] 11E76 (primary); 11E04, 11E81, 12G10, 16K20, 19D45 (secondary)

††journal: ??

1 Introduction

In this paper we study the connection between the Kato-Milne cohomology groups $H_{p}^{n+1}(F)$ over a field $F$ with $\operatorname{char}(F)=p$ for some prime integer $p$ , and homogeneous polynomial forms of degree $p$ over $F$ . The three main objectives of this work are:

Finding a number $n_{0}$ such that for any $n\geqslant n_{0}$ , $H_{p}^{n+1}(F)=0$ . 2. 2.

Finding an upper bound for the symbol length of $H_{p}^{2}(F)$ , which in turn provides an upper bound for the symbol length of $\prescript{}{p}{B}r(F)$ . 3. 3.

Finding a number $s$ such that any collection of $s$ inseparably linked decomposable differential forms in $H_{p}^{n+1}(F)$ are also separably linked.

1.1 The Kato-Milne Cohomology Groups

Given a prime number $p$ and a field $F$ of $\operatorname{char}(F)=p$ , we consider the space of absolute differential forms $\Omega_{F}^{1}$ , which is defined to be the $F$ -vector space generated by the symbols $da$ subject to the relations $d(a+b)=da+db$ and $d(ab)=adb+bda$ for any $a,b\in F$ . The space of $n$ -differential forms $\Omega_{F}^{n}$ for any positive integer $n$ is then defined by the $n$ -fold exterior power $\Omega_{F}^{n}=\bigwedge^{n}(\Omega_{F}^{1})$ , which is consequently an $F$ -vector space spanned by $da_{1}\wedge\ldots\wedge da_{n}$ , $a_{i}\in F$ . The derivation $d$ extends to an operator $d\,:\,\Omega_{F}^{n}\to\Omega_{F}^{n+1}$ by $d(a_{0}da_{1}\wedge\ldots\wedge da_{n})=da_{0}\wedge da_{1}\wedge\ldots\wedge da_{n}$ . We define $\Omega_{F}^{0}=F$ , $\Omega_{F}^{n}=0$ for $n<0$ , and $\Omega_{F}=\bigoplus_{n\geqslant 0}\Omega_{F}^{n}$ , the algebra of differential forms over $F$ with multiplication naturally defined by

[TABLE]

There exists a well-defined group homomorphism $\Omega_{F}^{n}\to\Omega_{F}^{n}/d\Omega_{F}^{n-1}$ , the Artin-Schreier map $\wp$ , which acts on decomposable differential forms as follows:

[TABLE]

The group $H_{p}^{n+1}(F)$ is defined to be $\operatorname{coker}(\wp)$ . By [15], in the case of $p=2$ , there exists an isomorphism

[TABLE]

where $\langle\langle\beta_{1},\dots,\beta_{n},\alpha]]$ is a quadratic $n$ -fold Pfister form.

By [14, Section 9.2], when $n=1$ , there exists an isomorphism

[TABLE]

where $[\alpha,\beta)_{p,F}$ is the degree $p$ cyclic $p$ -algebra

[TABLE]

In the special case of $p=2$ and $n=1$ , these cyclic $p$ -algebras are quaternion algebras $[\alpha,\beta)_{2,F}$ that can be identified with their norm forms which are quadratic 2-fold Pfister forms $\langle\langle\beta,\alpha]]$ (see [10, Corollary 12.2 (1)]).

1.2 $C_{m}$ and $\widetilde{C}_{p,m}$ Fields

A $C_{m}$ field is a field $F$ over which every homogeneous polynomial form of degree $d$ in more than $d^{m}$ variables is isotropic (i.e. has a nontrivial zero). It was suggested in [23, Chapter II, Section 4.5, Exercise 3 (b)] that if $F$ is a $C_{m}$ field with $\operatorname{char}(F)\neq p$ then for any $n\geqslant m$ , $H^{n+1}(F,\mu_{p}^{\otimes n})=0$ . This fact is known for $p=2$ because of the Milnor conjecture, proven in [24]. It was proven in [16] that for any prime $p>3$ , $C_{m}$ field $F$ with $\operatorname{char}(F)\neq p$ and $n\geqslant\lceil(m-2)\log_{2}(p)+1\rceil$ , we have $H^{n+1}(F,\mu_{p}^{\otimes n})=0$ . (The same result holds when $p=3$ for $n\geqslant\lceil(m-3)\log_{2}(3)+3\rceil$ .) The analogous statement for fields $F$ with $\operatorname{char}(F)=p$ is that if $F$ is a $C_{m}$ field then $H_{p}^{n+1}(F)=0$ for every $n\geqslant m$ . This is true, as stated in [23, Chapter II, Section 4.5, Exercise 3 (a)] and proven explicitly in [2]. It follows from the fact that $C_{m}$ fields $F$ have $p$ -rank at most $m$ , i.e. $[F:F^{p}]\leqslant p^{m}$ . We consider a somewhat different property of fields that avoids directly bounding their $p$ -rank. We say that a homogeneous polynomial form of degree $p$ over $F$ is $p$ -regular if there is no nonzero point where all the partial derivatives of order $p-1$ vanish. We denote by $u_{p}(F)$ the maximal dimension of an anisotropic $p$ -regular form over $F$ . We say $F$ is a $\widetilde{C}_{p,m}$ field if $u_{p}(F)\leqslant p^{m}$ . We prove that if $F$ is $\widetilde{C}_{p,m}$ then for any $n\geqslant\lceil(m-1)\log_{2}(p)\rceil+1$ , we have $H_{p}^{n+1}(F)=0$ . (See Section 2 for examples of $\widetilde{C}_{p,m}$ which are not $C_{m}$ .)

Remark 1.1.

Note that when $p=2$ , the notion of a $p$ -regular form coincides with nonsingular quadratic form, and $u_{p}(F)$ boils down to the $u$ -invariant $u(F)$ of $F$ . In this case, $\lceil(m-1)\log_{2}(p)\rceil+1=m$ , which recovers the known fact that when $u(F)\leqslant 2^{m}$ , we have $H_{2}^{m+1}(F)\cong I_{q}^{m+1}(F)/I_{q}^{m+2}(F)=0$ .

1.3 Symbol Length in $H_{p}^{2}(F)$

By [3, Theorem 30] (when $\operatorname{char}(F)=p$ ) and [21] (when $\operatorname{char}(F)\neq p$ and $F$ contains a primitive $p$ th root of unity), $\prescript{}{p}{B}r(F)$ is generated by cyclic algebras of degree $p$ . The symbol length of a class in $\prescript{}{p}{B}r(F)$ is the minimal number of cyclic algebras required in order to express this class as a tensor product of cyclic algebras. The symbol length of $\prescript{}{p}{B}r(F)$ is the supremum of the symbol length of all the classes in $\prescript{}{p}{B}r(F)$ . Recall that when $\operatorname{char}(F)=p$ , $\prescript{}{p}{B}r(F)\cong H_{p}^{2}(F)$ .

It was shown in [8, Corollary 3.3] that if the maximal dimension of an anisotropic form of degree $p$ over $F$ is $d$ then the symbol length of $\prescript{}{p}{B}r(F)$ is bounded from above by $\left\lceil\frac{d-1}{p}\right\rceil-1$ , providing a characteristic $p$ analogue to a similar result obtained in [20] in the case of $\operatorname{char}(F)\neq p$ . As a result, if $F$ is $C_{m}$ then $d\leqslant p^{m}$ and so this upper bound boils down to $p^{m-1}-1$ . However, the symbol length of $\prescript{}{p}{B}r(F)$ when $F$ is a $C_{m}$ field with $\operatorname{char}(F)=p$ is bounded from above by the $p$ -rank which is at most $m$ (see [3, Theorem 28]). We show that the forms discussed in [8] are actually $p$ -regular forms, which gives the upper bound $\left\lceil\frac{u_{p}(F)-1}{p}\right\rceil-1$ for the symbol length (which coincides with $\frac{u(F)}{2}-1$ when $p=2$ as in [8, Corollary 4.2]). In particular, if $F$ is $\widetilde{C}_{p,m}$ then the symbol length is bounded from above by $p^{m-1}-1$ . (This bound is in fact sharp for $p=2$ as proven in [8, Proposition 4.5].)

1.4 Separable and Inseparable Linkage

A differential form in $H_{p}^{n+1}(F)$ is called “decomposable” if it can be written as $\alpha\frac{d\beta_{1}}{\beta_{1}}\wedge\dots\wedge\frac{d\beta_{n}}{\beta_{n}}$ for some $\alpha\in F$ and $\beta_{1},\dots,\beta_{n}\in F^{\times}$ . We say that a collection of decomposable differential forms $\omega_{1},\dots,\omega_{m}$ in $H_{p}^{n+1}(F)$ are inseparably $\ell$ -linked if they can be written as

[TABLE]

such that $\beta_{1,k}=\dots=\beta_{m,k}$ for all $k\in\{1,\dots,\ell\}$ . We say they are separably $\ell$ -linked if they can be written in a similar way such that $\alpha_{1}=\dots=\alpha_{m}$ and $\beta_{1,k}=\dots=\beta_{m,k}$ for all $k\in\{1,\dots,\ell-1\}$ . By the identification of decomposable differential forms with quaternion algebras (when $p=2$ and $n=1$ ), cyclic $p$ -algebras (when $n=1$ ) and quadratic Pfister forms (when $p=2$ ), the notions of inseparable and separable linkage coincide with the previously defined notions of separable and inseparable linkages for these objects. In [9] it was proven that inseparable (1-)linkage of pairs of quaternion algebras implies separable (1-)linkage as well. A counterexample to the converse was given in [18]. These results extend naturally to Hurwitz algebras ([11]) and quadratic Pfister forms ([12, Corollary 2.1.4]). In [6, Corollary 5.4] it was shown that when $H_{2}^{n+2}(F)=0$ , separable $n$ -linkage and inseparable $n$ -linkage for pairs of quadratic $(n+1)$ -fold Pfister forms are equivalent. In [7] it was proven that inseparable $(1-)$ linkage for pairs of cyclic $p$ -algebras of degree $p$ implies separable $(1-)$ linkage as well, and that the converse is not necessarily true. It follows immediately that if two decomposable differential forms in $H_{p}^{n+1}(F)$ are inseparably $k$ -linked then they are also separably $k$ -linked. In this paper we generalize this statement for larger collections of forms: every collection of $1+\sum_{i=\ell}^{n}2^{i-1}$ inseparably $n$ -linked decomposable differential forms in $H_{p}^{n+1}(F)$ are also separably $\ell$ -linked. In particular, this means that if three octonion algebras share a biquadratic purely inseparable field extension of $F$ , then they also share a quaternion subalgebra.

2 Fields with bounded $u_{p}$ -invariant

There are examples in the literature of fields with $u(F)<2^{n+1}$ and unbounded $2$ -rank (see [22]). In particular, these fields are $\widetilde{C}_{2,m}$ but not $C_{m}$ .

Question 2.2.

Is there a similar construction of $\widetilde{C}_{p,m}$ fields which are not $C_{m}$ for prime numbers $p>2$ ?

The following construction gives an example of a field $F$ which is $\widetilde{C}_{p,0}$ but clearly not $C_{0}$ :

Example 2.3.

Let $K$ be a field of characteristic $p$ , $L=K(\lambda_{1},\dots,\lambda_{n})$ be the function field in $n$ algebraically independent variables ( $n$ can also be $\infty$ ), and $F=L^{\operatorname{sep}}$ the separable closure of $L$ . Then $F$ is $\widetilde{C}_{p,0}$ and not $C_{0}$ .

Proof.

Clearly it is not $C_{0}$ because its $p$ -rank is at least $n$ . To show that $F$ is $\widetilde{C}_{p,0}$ it is enough to show that every $p$ -regular form $\varphi(x_{1},\dots,x_{m})$ of dimension $m\geqslant 1$ over $F$ is isotropic. Let $\varphi(x_{1},\dots,x_{m})$ be a $p$ -regular form of dimension $m$ over $F$ . Since there are no $p$ -regular forms of dimension 1, $m>1$ . Since $\varphi$ is $p$ -regular, there exists a term with mixed variables and nonzero coefficient. Without loss of generality, assume the power of $x_{1}$ in this term is $d$ where $1\leqslant d\leqslant p-1$ . Write $\varphi$ as a polynomial in $x_{1}$ and coefficients in $F[x_{2},\dots,x_{m}]$ : $\varphi=c_{p}x_{1}^{p}+\dots+c_{1}x_{1}+c_{0}$ . The coefficient $c_{d}$ is a nonzero homogeneous polynomial form of degree $p-d$ in $m-1$ variables. Since it is nonzero, we have $c_{d}(a_{2},\dots,a_{m})\neq 0$ for some $a_{2},\dots,a_{m}\in F$ , not all zero. Without loss of generality, assume that $a_{2}\neq 0$ , which means we could assume $a_{2}=1$ . Then $c_{d}(x_{2},a_{3}x_{2},\dots,a_{m}x_{2})$ is a nonzero 1-dimensional form. Hence $\varphi(x_{1},x_{2},a_{3}x_{2},\dots,a_{m}x_{2})$ is a nondiagonal 2-dimensional form of degree $p$ . We shall explain now why this form must be isotropic: Suppose it is anisotropic. Consider the polynomial $\varphi(x_{1},1,a_{3},\dots,a_{m})$ . This is a polynomial of degree $\leqslant p$ and at least $d$ . If its degree is smaller than $p$ then since $F$ is separably closed, the polynomial decomposes into linear factors over $F$ , and then it has a root in $F$ , which means that $\varphi$ is isotropic. Assume the degree is $p$ . Since it is of degree $p$ , by [19, Chapter V, Corollary 6.2] this field extension must be either separable or purely inseparable. It cannot be purely inseparable because it has a nonzero term besides the degree $p$ and [math] terms. Therefore it must be separable, but that contradicts the fact that $F$ is separably closed. ∎

One can construct fields $F$ with bounded $u_{p}(F)$ and infinite $p$ -rank in the following way:

Lemma 2.4.

Let $K$ be a field of characteristic $p$ with $p$ -rank $r$ . Then for any anisotropic $p$ -regular form $\varphi(x_{1},\dots,x_{n})$ of dimension $n$ , the function field $K(\varphi)=K(x_{1},\dots,x_{n}:\varphi(x_{1},\dots,x_{n})=0)$ has $p$ -rank $r+n-1$ . One can also take $r=\infty$ and then the $p$ -rank of $K(\varphi)$ is $\infty$ as well.

Proof.

The function field $K(\varphi)$ of $\varphi$ is a degree $p$ separable extension of the function field $K(x_{1},\dots,x_{n-1})$ in $n-1$ algebraically independent variables over $K$ . The $p$ -rank of $K(x_{1},\dots,x_{n-1})$ is $r+n-1$ by [13, Lemma 2.7.2]. By [13, Lemma 2.7.3] the $p$ -rank of $K(\varphi)$ is also $r+n-1$ . ∎

Corollary 2.5.

Let $K$ be a field of characteristic $p$ and $p$ -rank $r$ , and let $M$ be a positive integer. Then $K$ is a subfield of some field $F$ with $p$ -rank at least $r$ and $u_{p}(F)\leqslant M$ .

Proof.

This $F$ is taken to be the compositum of the function fields of all anisotropic $p$ -regular forms of dimension greater than $M$ . By the previous lemma, the $p$ -rank of $F$ is at least $r$ . Clearly $u_{p}(F)\leqslant M$ . ∎

The last corollary provides examples of fields $F$ which are $\widetilde{C}_{p,m}$ by taking $M=p^{m}$ . The fact that if $F$ is $C_{m}$ then the field of Laurent series $F((\lambda))$ over $F$ is $C_{m+1}$ does not hold for $\widetilde{C}_{p,m}$ fields in general. For example, if one takes one of the fields constructed in [22] with $u(F)=2^{m}<\hat{u}(F)$ , then $F$ is $\widetilde{C}_{2,m}$ . However, by [5, Corollary 2.10], $u(F((\lambda)))=2\hat{u}(F)>2^{m+1}$ , hence $F((\lambda))$ is not $\widetilde{C}_{2,m+1}$ .

3 Symbol Length and $p$ -Regular forms

In this section we describe certain properties of $p$ -regular forms and make a note on the symbol length of classes in $\prescript{}{p}{B}r(F)$ when $F$ is a field of characteristic $p$ with bounded $u_{p}(F)$ .

Let $p$ be a prime integer and $F$ be a field of characteristic $p$ . Let $V=Fv_{1}+\dots+Fv_{m}$ be an $m$ -dimensional $F$ -vector space. A map $\varphi:V\rightarrow F$ is called a homogeneous polynomial form of degree $p$ if it satisfies

[TABLE]

for any $a_{1},\dots,a_{m}\in F$ where $c_{i_{1},\dots,i_{m}}$ are constants in $F$ . We say that $\varphi$ is isotropic if there exists a nonzero $v$ in $V$ such that $\varphi(v)=0$ . Otherwise $\varphi$ is anisotropic. We say $\varphi$ is $p$ -regular if there is no $v\in V\setminus\{0\}$ for which all the order $p-1$ partial derivatives of $\varphi$ vanish. The nonexistence of such points does not depend on the choice of basis. In the special case of $p=2$ , this notion coincides with nonsingularity. In particular, diagonal forms of degree $p$ over $F$ are not $p$ -regular. Given a homogeneous polynomial form $\varphi:V\rightarrow F$ , we can consider the scalar extension $\varphi_{L}$ of $\varphi$ from $F$ to $L$ .

Lemma 3.6.

Given a field extension $L/F$ , if $\varphi_{L}$ is a $p$ -regular form for some homogeneous polynomial form $\varphi$ of degree $p$ over $F$ then $\varphi$ is $p$ -regular as well.

Proof.

Assume the contrary, that $\varphi$ is not $p$ -regular, i.e. there exists $v\neq 0$ such that all the order $p-1$ partial derivatives of $\varphi$ vanish. Since the partial derivatives do not change under scalar extension, all the partial derivatives of $\varphi_{L}$ vanish at $v$ , which means that $\varphi_{L}$ is not $p$ -regular. ∎

Definition 3.7.

Given homogeneous polynomial forms $\varphi:V\rightarrow F$ and $\phi:W\rightarrow F$ of degree $p$ , we define the direct sum $\varphi\perp\phi$ to be the homogeneous polynomial form $\psi:V\oplus W\rightarrow F$ defined by $\psi(v+w)=\varphi(v)+\phi(w)$ for any $v\in V$ and $w\in W$ .

Lemma 3.8.

The form $\varphi\perp\phi$ is $p$ -regular if and only if both $\varphi$ and $\phi$ are $p$ -regular.

Proof.

If $\varphi$ is not $p$ -regular, there exists a nonzero $v\in V$ such that all the order $p-1$ partial derivatives of $\varphi$ vanish. Then all the order $p-1$ partial derivatives of $\varphi\perp\phi$ vanish at the point $v\oplus 0$ .

In the opposite direction, if $\varphi\perp\phi$ is not $p$ -regular, then there exists a nonzero $v\oplus w$ where all the order $p-1$ partial derivatives of $\varphi\perp\phi$ vanish. Without loss of generality, assume $v\neq 0$ . Then all the order $p-1$ partial derivatives of $\varphi\perp\phi$ vanish at $v$ . Since these derivatives are equal to the derivatives of $\varphi$ at $v$ , $\varphi$ is not $p$ -regular. ∎

Lemma 3.9.

Given a separable field extension $L/F$ of degree $p$ , the norm form $N:L\rightarrow F$ is a homogeneous polynomial form of degree $p$ . This form is $p$ -regular.

Proof.

By Lemma 3.6 it is enough to show that the scalar extension of $N$ to the algebraic closure $\overline{F}$ of $F$ is $p$ -regular. Now, $L\otimes_{F}\overline{F}$ is $\underbrace{\overline{F}\times\dots\times\overline{F}}_{p\enspace\text{times}}$ and can be identified with the $p\times p$ diagonal matrices with entries in $\overline{F}$ . Therefore we have

[TABLE]

The latter is clearly $p$ -regular. ∎

Remark 3.10.

If $\varphi:V\rightarrow F$ is $p$ -regular then for any nonzero scalar $c\in F$ , $c\varphi$ defined by $(c\varphi)(v)=c\varphi(v)$ for any $v\in V$ is also $p$ -regular.

Lemma 3.11.

Given a prime number $p$ and a field $F$ with $\operatorname{char}(F)\geqslant p$ or [math], the homogeneous polynomial form

[TABLE]

over the two-dimensional space $V=Fv_{1}+Fv_{2}$ is $p$ -regular for any $\alpha\in F$ .

Proof.

It is enough to note that the partial derivative obtained by differentiating $p-1$ times with respect to $a_{2}$ is $(p-1)!a_{1}$ and the partial derivative obtained by differentiating $p-2$ times with respect to $a_{2}$ and once with respect to $a_{1}$ is $-(p-1)!a_{2}$ . Therefore the only point where all the order $p-1$ partial derivatives vanish is $(0,0)$ and the form is $p$ -regular. ∎

In [8, Corollary 3.3] it was proven that the symbol length of a $p$ -algebra of exponent $p$ over $K$ is bounded by $\left\lceil\frac{d-1}{p}\right\rceil-1$ where $d$ is the maximal dimension of an anisotropic homogeneous polynomial form of degree $p$ over $K$ . Apparently $d$ can be replaced with $u_{p}(F)$ .

Theorem 3.12 (cf. [8, Theorem 3.2 and Theorem 4.1]).

Let $p$ be a prime integer and let $F$ be a field with $\operatorname{char}(F)=p$ and finite $u_{p}(F)$ . Then every two tensor products $A=\bigotimes_{i=1}^{m}[\alpha_{i},\beta_{i})_{p,F}$ and $B=\bigotimes_{i=1}^{\ell}[\gamma_{i},\delta_{i})_{p,F}$ with $(m+\ell)p\geqslant u_{p}(F)-1$ can be changed such that $\alpha_{1}=\gamma_{1}$ .

Proof.

It is enough to show that the homogeneous polynomial form considered in the proof of [8, Theorem 3.2] is $p$ -regular. This form $\varphi$ is the direct sum $\varphi^{\prime}\perp\phi_{1}\perp\dots\perp\phi_{m}\perp\psi_{1}\perp\dots\perp\psi_{\ell}$ where $\varphi^{\prime}:F\times F\rightarrow F$ is defined by $\varphi^{\prime}(a,b)=(\sum_{i=1}^{m}(\alpha_{i})-\sum_{i=1}^{\ell}(\gamma_{i}))a^{p}-a^{p-1}b+b^{p}$ , for each $i\in\{1,\dots,m\}$ , $\phi_{i}$ is $\beta_{i}N_{F[x:x^{p}-x=\alpha_{i}]/F}$ , and for each $i\in\{1,\dots,\ell\}$ , $\psi_{i}$ is $\gamma_{i}N_{F[x:x^{p}-x=\delta_{i}]/f}$ . By Lemma 3.9 and Remark 3.10, the forms $\phi_{1},\dots,\phi_{m},\psi_{1},\dots,\psi_{\ell}$ are $p$ -regular. By Lemma 3.11, $\varphi^{\prime}$ is also $p$ -regular. Consequently, the form $\varphi$ is $p$ -regular as direct sum of $p$ -regular forms by Lemma 3.8. ∎

Corollary 3.13 (cf. [8, Corollary 3.3 and Corollary 4.2]).

Let $p$ be a prime integer and let $F$ be a field with $\operatorname{char}(F)=p$ and finite $u_{p}(F)$ . Then the symbol length in $H_{p}^{2}(F)$ is bounded from above by $\left\lceil\frac{u_{p}(F)-1}{p}\right\rceil-1$ .

Proof.

It follows from Theorem 3.12 in the same manner [8, Corollary 3.3] follows from [8, Theorem 3.2]. ∎

Corollary 3.14.

Let $p$ be a prime integer and let $F$ be a $\widetilde{C}_{p,m}$ field with $\operatorname{char}(F)=p$ . Then the symbol length in $H_{p}^{2}(F)$ is bounded from above by $p^{m-1}-1$ .

Proof.

An immediate result of the previous corollary, given that for a $\widetilde{C}_{p,m}$ field $F$ , $u_{p}(F)\leqslant p^{m}$ . ∎

4 Symbol Length and $p$ -Rank

By [3, Chapter 7, Theorem 28], the $p$ -rank of $F$ is an upper bound for the symbol length of $H_{p}^{2}(F)$ . In fact, it can be shown that if the $p$ -rank of $F$ is a finite integer $m$ then the symbol length of $H_{p}^{n+1}(F)$ is bounded from above by $\binom{m}{n}$ for any positive integer $n$ .

Remark 4.15.

If $F$ is $C_{m}$ then its $p$ -rank is $\leqslant m$ .

Proof.

Suppose $F$ is $C_{m}$ and that its $p$ -rank is $n$ . Then $F=F^{p}v_{1}\oplus\dots\oplus F^{p}v_{p^{n}}$ where $v_{1},\dots,v_{p^{n}}$ are linearly independent over $F^{p}$ . Therefore the form $\varphi(a_{1},\dots,a_{p^{n}})=v_{1}a_{1}^{p}+\dots+v_{p^{n}}a_{p^{n}}^{p}$ is anisotropic. Since the dimension of $\varphi$ is $p^{n}$ , $m$ must be $\geqslant n$ . ∎

For a $C_{m}$ field (compared to $\widetilde{C}_{p,m}$ ) $F$ , its $p$ -rank ( $\leqslant m$ ) provides a better upper bound for the symbol length of $H_{p}^{2}(F)$ than the upper bound given in Corollary 3.14 ( $p^{m-1}-1$ ). The following proposition shows that there exist cases where the symbol length is actually equal to the $p$ -rank:

Proposition 4.16.

Let $K=F((\beta_{1}))\dots((\beta_{n}))$ be the field of iterated Laurent series in $n$ variables over a perfect field $F$ of characteristic $p$ . Let $L/F$ be a $(\mathbb{Z}/p\mathbb{Z})^{n}$ -Galois field extension given by $L=F[\wp^{-1}(\alpha_{1}),\dots,\wp^{-1}(\alpha_{n})]$ . Then the symbol length of the $p$ -algebra

[TABLE]

is equal to the $p$ -rank of $K$ .

Proof.

The $p$ -rank of $K$ in the proposition above is $n$ by [13, Chapter 2, Lemma 2.7.2]. The $p$ -algebra $D$ is a generic abelian crossed product with maximal $(\mathbb{Z}/p\mathbb{Z})^{n}$ -Galois subfield $K[\wp^{-1}(\alpha_{1}),\dots,\wp^{-1}(\alpha_{n})]$ , hence it is a division algebra (see [4]) and its symbol length is exactly $n$ . ∎

Remark 4.17.

In order to construct a Galois extension satisfying the conditions of Proposition 4.16, take $\alpha_{1},\ldots,\alpha_{n}$ to be algebraically independent variables over $\mathbb{F}_{p}$ . Let $F$ be the perfect closure of $\mathbb{F}_{p}(\alpha_{1},\ldots,\alpha_{n})$ . Then each field extension $F[\wp^{-1}(\alpha_{i})]$ is $(\mathbb{Z}/p\mathbb{Z})$ -Galois and as a set they are mutually linearly independent. Therefore $L=F[\wp^{-1}(\alpha_{1}),\ldots,\wp^{-1}(\alpha_{n})]$ is a $(\mathbb{Z}/p\mathbb{Z})^{n}$ -Galois extension of the perfect field $F$ .

The following example presents a $C_{m}$ field with $p$ -rank $n$ such that $m=2n+1$ :

Example 4.18.

Let $F$ be the perfect closure of the function field $\mathbb{F}_{p}(\alpha_{1},\dots,\alpha_{n})$ as in remark 4.17. Let $K=F((\beta_{1}))\dots((\beta_{n}))$ be the field of iterated Laurent series in $n$ variables over $F$ . As mentioned above, the $p$ -rank of $K$ is $n$ , and the symbol length of the algebra $[\alpha_{1},\beta_{1})_{p,F}\otimes\dots\otimes[\alpha_{n},\beta_{n})_{p,F}$ is $n$ . The field $\mathbb{F}_{p}(\alpha_{1},\dots,\alpha_{n})$ is a $C_{n+1}$ field, and hence so is its perfect closure $F$ . Consequently, $K$ is a $C_{2n+1}$ field.

5 Class-Preserving Modifications of Decomposable Differential Forms

In this section we study the class-preserving modifications of decomposable differential forms in $H_{p}^{n+1}(F)$ . These modifications will be used in proving the main results of the following sections. In the special case of $p=2$ they coincide with the known modifications of quadratic Pfister forms (see [1]).

Lemma 5.19.

Let $\omega=\alpha\frac{d\beta_{1}}{\beta_{1}}\wedge\ldots\wedge\frac{d\beta_{n}}{\beta_{n}}$ be a form in $H_{p}^{n+1}(F)$ . Then:

$(a)$

For any $i\in\{1,\dots,n\}$ , $\omega=(\alpha+\beta_{i})\frac{d\beta_{1}}{\beta_{1}}\wedge\ldots\wedge\frac{d\beta_{n}}{\beta_{n}}.$ 2. $(b)$

For any $i\in\{1,\dots,n\}$ and nonzero $f\in F[\wp^{-1}(\alpha)]$ , if $\operatorname{N}_{F[\wp^{-1}(\alpha)]/F}(f)=0$ then $\omega=0$ . Otherwise, $\omega=\alpha\frac{d\beta_{1}}{\beta_{1}}\wedge\ldots\wedge\frac{d(\operatorname{N}_{F[\wp^{-1}(\alpha)]/F}(f)\beta_{i})}{\operatorname{N}_{F[\wp^{-1}(\alpha)]/F}(f)\beta_{i}}\wedge\dots\wedge\frac{d\beta_{n}}{\beta_{n}}.$ 3. $(c)$

For any $i\in\{1,\dots,n\}$ and $\gamma\in F$ , if $\beta_{i}+\gamma^{p}=0$ then $\omega=0$ . Otherwise, there exists some $\alpha^{\prime}\in F$ such that $\omega=\alpha^{\prime}\frac{d\beta_{1}}{\beta_{1}}\wedge\ldots\wedge\frac{d(\beta_{i}+\gamma^{p})}{\beta_{i}+\gamma^{p}}\wedge\dots\wedge\frac{d\beta_{n}}{\beta_{n}}.$ 4. $(d)$

For any distinct $i,j\in\{1,\dots,n\}$ ,

[TABLE] 5. $(e)$

For any distinct $i,j\in\{1,\dots,n\}$ , if $\beta_{i}+\beta_{j}=0$ then $\omega=0$ . Otherwise,

[TABLE] 6. $(f)$

For any distinct $i,j\in\{1,\dots,n\}$ and $f\in F[\wp^{-1}(\alpha)]$ , if $\beta_{i}+\operatorname{N}_{F[\wp^{-1}(\alpha)]/F}(f)=0$ then $\omega=0$ . Otherwise,

[TABLE]

Proof.

Parts $(a),(b)$ and $(c)$ are elementary and follow immediately from the equivalent statements for cyclic $p$ -algebras (see [8, Lemma 2.2]). Part $(d)$ follows from the linearity of logarithmic differential forms: $\frac{d(\beta_{i}\beta_{j})}{\beta_{i}\beta_{j}}=\frac{d\beta_{i}}{\beta_{i}}+\frac{d\beta_{j}}{\beta_{j}}$ .

For part $(e)$ , if $\beta_{i}=-\beta_{j}$ then $d\beta_{i}\wedge d\beta_{j}=0$ and so $\omega=0$ . Otherwise, it is enough to show that

[TABLE]

To see this, notice that $d(\beta_{i}+\beta_{j})\wedge d(\beta_{i}^{-1}\beta_{j})=(\beta_{i}^{-1}+\beta_{i}^{-2}\beta_{j})d\beta_{i}\wedge d\beta_{j}$ and divide both sides by $(\beta_{i}+\beta_{j})(\beta_{i}^{-1}\beta_{j})$ .

For $(f)$ , recall that for any $a\in F$ and $b\in F^{\times}$ , $[a,b)_{p,F}$ is split if and only if $b$ is a norm in the étale extension $F[\wp^{-1}(a)]/F$ , and otherwise it is a division algebra. If $\beta_{i}+\operatorname{N}_{F[\wp^{-1}(\alpha)]/F}(f)=0$ then the algebra $[\alpha,\beta_{i})_{p,F}=0$ by the norm condition, and so $\omega=0$ . Otherwise, consider the form

[TABLE]

By $(d)$ , it is equal to the sum of $\omega$ and the form

[TABLE]

If $\operatorname{N}_{F[\wp^{-1}(\alpha)]/F}(f)=0$ then this form is clearly zero. Otherwise, by $(b)$ it is equal to

[TABLE]

which is zero by $(c)$ . ∎

Part (b) of Lemma 5.19 shows that $\beta_{n}$ can be replaced by any nonzero element represented by the $p$ -regular form $\varphi:F[\wp^{-1}(\alpha)]\to F$ with $f\mapsto\beta_{n}\operatorname{N}_{F[\wp^{-1}(\alpha)]/F}(f)$ . In the following proposition we present a $p$ -regular form of larger dimension whose values can also alter the last slot (at the possible cost of changing some of the other inseparable slots).

Proposition 5.20.

Let $\alpha\in F$ and $\beta_{1},\dots,\beta_{n}\in F^{\times}$ , and write $\omega=\alpha\frac{d\beta_{1}}{\beta_{1}}\wedge\ldots\wedge\frac{d\beta_{n}}{\beta_{n}}$ for the corresponding form in $H_{p}^{n+1}(F)$ . For each $(d_{1},\dots,d_{n})\in\underbrace{\{0,1\}\times\dots\times\{0,1\}}_{n\enspace\text{times}}$ , let $V_{d_{1},\dots,d_{n}}$ be a copy of $F[\wp^{-1}(\alpha)]$ and $\varphi_{d_{1},\dots,d_{n}}:V_{d_{1},\dots,d_{n}}\rightarrow F$ be the homogeneous polynomial form of degree $p$ defined by $\varphi_{d_{1},\dots,d_{n}}(f)=\operatorname{N}_{F[\wp^{-1}(\alpha)]/F}(f)\cdot\beta_{1}^{d_{1}}\cdot\ldots\cdot\beta_{n}^{d_{n}}$ . Write

[TABLE]

If there exists a nonzero $v$ such that $\varphi(v)=0$ then $\omega=0$ . Otherwise, for every nonzero $v\in V$ , there exist $\beta_{1}^{\prime},\dots,\beta_{n-1}^{\prime}\in F^{\times}$ such that :

[TABLE]

Remark 5.21.

To get a feel for the form $(\varphi,V)$ , note that if we set $N=N_{F[\wp^{-1}(\alpha)]/F}$ then for $n=1$ , $(\varphi,V)=(\varphi_{1},V_{1}):F[\wp^{-1}(\alpha)]\to V$ with $x\mapsto N(x)\beta_{1}$ . When $n=2$ , $(\varphi,V):F[\wp^{-1}(\alpha)]^{\times 3}\to F$ with $(x,y,z)\mapsto N(x)\beta_{1}+N(y)\beta_{2}+N(z)\beta_{1}\beta_{2}$ .

Proof of Proposition 5.2.

By induction on $n$ . The case of $n=1$ holds by the analogy to cyclic $p$ -algebras. Assume it holds for a certain $n-1$ . We shall show it holds also for $n$ . The vector space $V$ decomposes as $V_{0}\oplus V_{1}$ where $V_{0}=\mathop{\mathchoice{\vbox{ \hbox{\scalebox{2.1}{$ \displaystyle\perp $}} }}{\vbox{ \hbox{\scalebox{1.5}{$ \textstyle\perp $}} }}{\vbox{ \hbox{\scalebox{1.5}{$ \scriptstyle\perp $}} }}{\vbox{ \hbox{\scalebox{1.5}{$ \scriptscriptstyle\perp $}} }}}\displaylimits_{0\leqslant d_{1},\dots,d_{n-1}\leqslant 1}V_{d_{1},\dots,d_{n-1},0}$ and $V_{1}=\mathop{\mathchoice{\vbox{ \hbox{\scalebox{2.1}{$ \displaystyle\perp $}} }}{\vbox{ \hbox{\scalebox{1.5}{$ \textstyle\perp $}} }}{\vbox{ \hbox{\scalebox{1.5}{$ \scriptstyle\perp $}} }}{\vbox{ \hbox{\scalebox{1.5}{$ \scriptscriptstyle\perp $}} }}}\displaylimits_{0\leqslant d_{1},\dots,d_{n-1}\leqslant 1}V_{d_{1},\dots,d_{n-1},1}$ . The latter decomposes as $V_{1}=V_{0,\dots,0,1}+V_{1}^{\prime}$ . There is a natural isomorphism $\tau:V_{1}^{\prime}\rightarrow V_{0}$ identifying each $V_{d_{1},\dots,d_{n-1},1}$ with $V_{d_{1},\dots,d_{n-1},0}$ . Under this homomorphism, we have $\varphi(v)=\beta_{n}\varphi(\tau(v))$ for any $v\in V_{1}^{\prime}$ .

Let $v$ be a nonzero vector in $V$ . Then $v=v_{0}+v_{1}^{\prime}+v_{0,1}$ where $v_{0}\in V_{0}$ , $v_{1}^{\prime}\in V_{1}^{\prime}$ and $v_{0,1}\in V_{0,\dots,0,1}(\cong F[\wp^{-1}(\alpha)])$ . Note that $\varphi(v)=\varphi(v_{0})+\varphi(v_{1}^{\prime})+\varphi(v_{0,1})=\varphi(v_{0})+\beta_{n}(\varphi(\tau(v_{1}))+\operatorname{N}_{F[\wp^{-1}(\alpha)]/F}(v_{0,1}))$ . Write $v_{1}=v_{1}^{\prime}+v_{0,1}$ . If $v_{1}^{\prime}=0$ and $v_{0,1}=0$ then the statement follows immediately from the induction hypothesis. If $v_{1}^{\prime}=0$ and $v_{0,1}\neq 0$ then the statement follows from the induction hypothesis and Lemma 5.19 $(e)$ .

Assume $v_{1}^{\prime}\neq 0$ . If $\varphi(\tau(v_{1}^{\prime}))=0$ then by the induction hypothesis, $\omega=0$ . Otherwise, by the induction hypothesis, there exist $\beta_{1}^{\prime},\dots,\beta_{n-2}^{\prime}$ such that

[TABLE]

By Lemma 5.19 $(f)$ , if $\varphi(\tau(v_{1}^{\prime}))+\operatorname{N}_{F[\wp^{-1}(\alpha)]/F}(v_{0,1})=0$ then $\omega=0$ , and otherwise we have

[TABLE]

Consequently

[TABLE]

If $v_{0}=0$ , this completes the picture. Assume $v_{0}\neq 0$ . By the assumption, there exist $\beta_{1}^{\prime\prime},\dots,\beta_{n-2}^{\prime\prime}$ such that

[TABLE]

By Lemma 5.19 $(e)$ , if $\varphi(v_{0})+\varphi(v_{1})=0$ then $\omega=0$ , and otherwise

[TABLE]

and since $\varphi(v)=\varphi(v_{0})+\varphi(v_{1})$ , this proves the statement. ∎

Corollary 5.22.

Using the same setting as Proposition 5.20, write

[TABLE]

Then for every nonzero $v_{1}\in V_{1}$ , assuming $\omega\neq 0$ , we have

[TABLE]

Proof.

The vector space $V_{1}$ decomposes as $V_{1}^{\prime}\oplus V_{0,\dots,0,1}$ where $V_{1}^{\prime}$ is the direct sum of all $V_{d_{1},\dots,d_{n-1},1}$ with $(d_{1},\dots,d_{n-1})\neq(0,\dots,0)$ . Take $\tau$ to be the natural isomorphism from $V_{1}^{\prime}$ to $V_{0}$ . Let $v_{1}$ be a nonzero element in $V_{1}$ . It can therefore be written as $v_{1}=v_{1}^{\prime}+v_{0,1}$ where $v_{1}^{\prime}\in V_{1}^{\prime}$ and $v_{0,1}\in V_{0,\dots,0,1}(\cong F[\wp^{-1}(\alpha)])$ . By the previous proposition

[TABLE]

for some $\beta_{1}^{\prime},\dots,\beta_{n-2}^{\prime}\in F^{\times}$ . Consequently,

[TABLE]

By Lemma 5.19 $(f)$ we have

[TABLE]

and so

[TABLE]

∎

Corollary 5.23.

Using the same setting as Proposition 5.20, let $\ell$ be an integer between $1$ and $n$ . Write

[TABLE]

Then for every nonzero $v\in W$ , assuming $\omega\neq 0$ , there exist $\beta_{\ell}^{\prime},\dots,\beta_{n-1}^{\prime}\in F$ such that

[TABLE]

Proof.

The vector space $W$ decomposes as $W_{n}\oplus W_{n-1}\oplus\dots\oplus W_{\ell}$ such that for each $k$ between $\ell$ and $n$ ,

[TABLE]

Let $v$ be a nonzero vector in $W$ . Then $v$ can be written accordingly as $v=v_{n}+\dots+v_{\ell}$ where each $v_{k}$ belongs to $W_{k}$ . For each $k\in\{\ell,\dots,n\}$ , by the previous lemma

[TABLE]

Therefore

[TABLE]

Then by Proposition 5.19 $(e)$ we can change the last term to be $\frac{d(\varphi(v_{\ell})+\dots+\varphi(v_{n}))}{\varphi(v_{\ell})+\dots+\varphi(v_{n})}$ at the cost of possibly changing the slots $\ell$ to $n-1$ . ∎

6 Linkage of Decomposable Differential Forms

Theorem 6.24.

Given a field $F$ of $\operatorname{char}(F)=p$ , a positive integer $n$ and an integer $\ell\in\{1,\dots,n\}$ , every collection of $1+\sum_{i=\ell}^{n}2^{i-1}$ inseparably $n$ -linked decomposable differential forms in $H_{p}^{n+1}(F)$ are separably $\ell$ -linked as well.

Proof.

Write $m=2^{n-1}+\dots+2^{\ell-1}$ . Let $\left\{\alpha_{i}\frac{d\beta_{1}}{\beta_{1}}\wedge\dots\wedge\frac{d\beta_{n}}{\beta_{n}}:i\in\{0,\dots,m\}\right\}$ be the collection of $m+1$ inseparably $n$ -linked decomposable differential forms in $H_{p}^{n+1}(F)$ under discussion. The number of $n$ -tuples $(d_{1},\dots,d_{n})$ with $(d_{\ell},\dots,d_{n})\neq(0,\dots,0)$ is $m$ . We denote arbitrarily the elements of the set $\{\beta_{1}^{d_{1}}\cdot\ldots\cdot\beta_{n}^{d_{n}}:0\leqslant d_{1},\dots,d_{n}\leqslant 1,(d_{\ell},\dots,d_{n})\neq(0,\dots,0)\}$ by $\gamma_{1},\dots,\gamma_{m}$ (with possible repetitions). Recall that the norm of an element $x+\lambda y$ in $F[\lambda:\lambda^{p}-\lambda=\alpha]$ is $x^{p}-xy^{p-1}+y^{p}\alpha$ . Therefore by Corollary 5.23, for every $i\in\{0,\dots,m\}$ and any choice of $x_{i,1},y_{i,1},\dots,x_{i,m},y_{i,m}\in F$ with $(x_{i,1},\dots,y_{i,m})\neq(0,\dots,0)$ we can change the form $\alpha_{i}\frac{d\beta_{1}}{\beta_{1}}\wedge\dots\wedge\frac{d\beta_{n}}{\beta_{n}}$ to $\alpha_{i}\frac{d\beta_{1}}{\beta_{1}}\wedge\dots\wedge\frac{d\beta_{\ell-1}}{\beta_{\ell-1}}\wedge\frac{db_{i,ell}}{b_{i,ell}}\wedge\dots\wedge\frac{db_{i,n-1}}{b_{i,n-1}}\wedge\frac{d\delta_{i}}{\delta_{i}}$ where

[TABLE]

for some $b_{i,\ell},\dots,b_{i,n-1}\in F$ .

Therefore, in order to show that the forms in the collection are separably $\ell$ -linked, it is enough to show that the following system of $m$ equations in $2m(m+1)$ variables has a solution:

[TABLE]

If we take $x_{0,i}=x_{1,i}=\dots=x_{m,i}$ , $y_{0,i}=1$ and $y_{i,i}=0$ and $y_{i,j}=1$ for all $i,j\in\{1,\dots,m\}$ with $i\neq j$ , then the $i$ th equation in this system becomes a linear equation in one variable $x_{0,i}$ (whose coefficient is $\gamma_{i}$ ). This system therefore has a solution. ∎

Corollary 6.25.

Given a field $F$ of $\operatorname{char}(F)=2$ , a positive integer $n$ and an integer $\ell\in\{1,\dots,n\}$ , every collection of $1+\sum_{i=\ell}^{n}2^{i-1}$ inseparably $n$ -linked quadratic $(n+1)$ -fold Pfister forms are also separably $\ell$ -linked.

Proof.

Follows Theorem 6.24 and the identification of quadratic Pfister forms with decomposable differential forms appearing in [15]. ∎

By the identification of octonion algebras with their 3-fold Pfister norm forms ([17, Theorem 33.19]), we conclude that if three octonion $F$ -algebras share a biquadratic purely inseparable field extension of $F$ then they also share a quaternion subalgebra. (Recall that when $\operatorname{char}(F)=2$ , an octonion algebra $A$ over $F$ is of the form $Q+Qz$ where $Q=[\alpha,\beta)_{2,F}$ is a quaternion algebra, $z^{2}=\gamma$ and $z\ell=\ell^{\sigma}z$ for every $\ell\in Q$ where $\sigma$ is the canonical involution on $Q$ , $\alpha\in F$ and $\beta,\gamma\in F^{\times}$ . In particular, when $A$ is a division algebra, $F[\sqrt{\beta},\sqrt{\gamma}]$ is a subfield of $A$ and a biquadratic purely inseparable field extension of $F$ .)

It is not known to the authors if the sizes of the collections mentioned in Theorem 6.24 are sharp in the sense that larger collections of inseparably $n$ -linked differential forms in $H_{p}^{n+1}(F)$ need not be separably $\ell$ -linked. Even the very special case of three inseparably linked quaternion algebras would be very interesting, in either direction.

Question 6.26.

Are every three inseparably linked quaternion algebras also separably linked?

7 Vanishing Cohomology Groups

One can use Proposition 5.20 to show that for fields $F$ with finite $u_{p}(F)$ the cohomology groups $H_{p}^{n+1}(F)$ vanish from a certain point and on. See section 1.2 for the definition of $u_{p}(F)$ .

Theorem 7.27.

Given a field $F$ with $\operatorname{char}(F)=p$ , for any $n$ with $(2^{n}-1)p\geqslant u_{p}(F)$ , $H^{n+1}_{p}(F)=0$ .

Proof.

Let $\omega=\alpha\frac{d\beta_{1}}{\beta_{1}}\wedge\ldots\wedge\frac{d\beta_{n-1}}{\beta_{n-1}}\wedge\frac{d\beta_{n}}{\beta_{n}}$ be an arbitrary decomposable form in $H_{p}^{n+1}(F)$ . Consider the direct sum $\Phi$ of the form $\varphi$ from Proposition 5.20 and the two-dimensional form $\psi(x,y)=\alpha x^{p}-x^{p-1}y+y^{p}$ . Note that by Lemmas 3.6, 3.8 and 3.11, $\Phi$ is $p$ -regular. If $(2^{n}-1)p+1\geqslant u_{p}(F)$ then ${\operatorname{dim}}(\Phi)=(2^{n}-1)p+2$ , hence $\Phi$ is isotropic, which means that there exist $(x_{0},y_{0},v_{0})\neq 0$ such that $\psi(x_{0},y_{0})+\varphi(v_{0})=0$ . If $v_{0}=0$ then it means $\alpha\in\wp(F)$ , and therefore $\omega$ is trivial. If $\varphi$ is isotropic then $\omega$ is trivial by Proposition 5.20. Otherwise, $\varphi(v_{0})\neq 0$ . By proposition 5.20, $\omega=\alpha\frac{d\beta_{1}^{\prime}}{\beta_{1}^{\prime}}\wedge\dots\wedge\frac{d\beta_{n-1}^{\prime}}{\beta_{n-1}^{\prime}}\wedge\frac{d\varphi(v_{0})}{\varphi(v_{0})}$ for some $\beta_{1}^{\prime},\dots,\beta_{n-1}^{\prime}\in F$ . If $x_{0}=0$ then $\Phi(0,y_{0},v_{0})=y_{0}^{p}+\varphi(v_{0})=0$ , and so $\varphi(v_{0})=(-y_{0})^{p}$ , which means $\omega$ is trivial. If $x_{0}\neq 0$ then $\Phi(1,\frac{y_{0}}{x_{0}},\frac{v_{0}}{x_{0}})=0$ as well. By Proposition 5.20, $\omega=\alpha\frac{d\beta_{1}^{\prime}}{\beta_{1}^{\prime}}\wedge\dots\wedge\frac{d\beta_{n-1}^{\prime}}{\beta_{n-1}^{\prime}}\wedge\frac{d\varphi(\frac{v_{0}}{x_{0}})}{\varphi(\frac{v_{0}}{x_{0}})}$ for some $\beta_{1}^{\prime},\dots,\beta_{n-1}^{\prime}\in F$ . Then, by Lemma 5.19 $(a)$ , we can add $\varphi(\frac{v_{0}}{x_{0}})$ to $\alpha$ , but then the coefficient is congruent to $\Phi(1,\frac{y_{0}}{x_{0}},\frac{v_{0}}{x_{0}})$ modulo $\wp(F)$ , and therefore $\omega$ is trivial. ∎

Corollary 7.28.

If $F$ is a $\widetilde{C}_{p,m}$ field with $\operatorname{char}(F)=p$ then for all $n\geqslant\lceil(m-1)\log_{2}(p)\rceil+1$ , $H_{p}^{n+1}(F)=0$ .

Proof.

Since $F$ is a $\widetilde{C}_{p,m}$ field, $u_{p}(F)\leqslant p^{m}$ . If $n\geqslant\lceil(m-1)\log_{2}(p)\rceil+1$ then $2^{n}\geqslant p^{m-1}+1$ , and so $2^{n}\cdot p\geqslant p^{m}+p$ , which means that $2^{n}\cdot p-p\geqslant p^{m}$ , and therefore $(2^{n}-1)p+1\geqslant p^{m}+1\geqslant u_{p}(F)+1$ . ∎

Example 7.29.

If we take $m=1$ then $\lceil(m-1)\log_{2}(p)\rceil+1=1$ , and therefore $H_{p}^{1+1}(F)=H_{p}^{2}(F)=0$ for $\widetilde{C}_{p,1}$ fields $F$ . The group $H_{p}^{2}(F)$ is isomorphic to $\prescript{}{p}{B}r(F)$ , which is generated by symbol algebras $[\alpha,\beta)_{p,F}$ . Another way of seeing that $\prescript{}{p}{B}r(F)$ is trivial for $\widetilde{C}_{p,1}$ is to consider an arbitrary generator $A=[\alpha,\beta)_{p,F}=F\langle x,y:x^{p}-x=\alpha,y^{p}=\beta,yxy^{-1}=x+1\rangle$ and to show that it must be split.

If $F[x]$ is not a field then this algebra is split, so suppose this is a field. Then consider the norm form $N:F[x]\rightarrow F$ . For any choice of $f\in F[x]^{\times}$ , the element $z=fy$ satisfies $z^{p}=N(f)\beta$ and $zxz^{-1}=x+1$ , so $A=[\alpha,N(F)\beta)_{p,F}$ . The element $w=x+z$ satisfies $w^{p}-w=\alpha+N(f)\beta$ (see [7, Lemma 3.1]). Now, consider the homogeneous polynomial form

[TABLE]

This form is $p$ -regular by Lemma 3.9, Remark 3.10 and Lemma 3.11. It is isotropic because its dimension is $p+2$ and $F$ is $\widetilde{C}_{p,1}$ , i.e. there exist $a_{0},b_{0}\in F$ and $f_{0}\in F[x]$ not all zero such that $\varphi(a_{0},b_{0},f_{0})=0$ . The case of $a_{0}=b_{0}=0$ is impossible because then $N(f_{0})=0$ which means that $F[x]$ is not a field, contrary to the assumption. If $a_{0}=0$ then the element $t=b_{0}+f_{0}y$ satisfies $t^{p}=0$ , which means $A$ is split. If $a_{0}\neq 0$ then the element $t=x+\frac{b_{0}}{a_{0}}+\frac{1}{a_{0}}f_{0}y$ satisfies $t^{p}-t=0$ , which again means $A$ is split.

Recall that for $p=2$ , the notion of $u_{p}(F)$ coincides with the $u$ -invariant of $F$ . Then the statement recovers the known fact that if $n$ satisfies $2^{n+1}>u(F)$ then $H_{2}^{n+1}(F)=0$ .

References

AB [92]

Roberto Aravire and Ricardo Baeza, Milnor’s $k$ -theory and quadratic forms over fields of characteristic two, Comm. Algebra 20 (1992), no. 4, 1087–1107. MR 1154404 (93a:11030)

AB [10]

Jon Kristinn Arason and R. Baeza, La dimension cohomologique des corps de type ${\bf C}_{\rm r}$ en caractéristique ${\bf p}$ , C. R. Math. Acad. Sci. Paris 348 (2010), no. 3-4, 125–126. MR 2600061

Alb [68]

A. Albert, Structure of algebras, Colloquium Publications, vol. 24, American Math. Soc., 1968.

AS [78]

S. A. Amitsur and D. Saltman, Generic Abelian crossed products and $p$ -algebras, J. Algebra 51 (1978), no. 1, 76–87. MR 0491789

Bae [82]

R. Baeza, Comparing $u$ -invariants of fields of characteristic $2$ , Bol. Soc. Brasil. Mat. 13 (1982), no. 1, 105–114. MR 692281

CGV [17]

Adam Chapman, Shira Gilat, and Uzi Vishne, Linkage of quadratic Pfister forms, Comm. Algebra 45 (2017), no. 12, 5212–5226. MR 3691360

Cha [15]

Adam Chapman, Common subfields of $p$ -algebras of prime degree, Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 4, 683–686. MR 3429179

Cha [17]

, Symbol length of $p$ -algebras of prime exponent, J. Algebra Appl. 16 (2017), no. 5, 1750136, 9.

Dra [75]

Peter Draxl, Über gemeinsame separabel-quadratische Zerfällungskörper von Quaternionenalgebren, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1975), no. 16, 251–259. MR 0480436

EKM [08]

Richard Elman, Nikita Karpenko, and Alexander Merkurjev, The algebraic and geometric theory of quadratic forms, American Mathematical Society Colloquium Publications, vol. 56, American Mathematical Society, Providence, RI, 2008. MR 2427530 (2009d:11062)

EV [05]

A. Elduque and O. Villa, A note on the linkage of Hurwitz algebras, Manuscripta Math. 117 (2005), no. 1, 105–110. MR 2142905 (2005m:11068)

Fai [06]

Frédéric Faivre, Liaison des formes de Pfister et corps de fonctions de quadriques en caractéristique $2$ , 2006, Thesis (Ph.D.)–Université de Franche-Comté.

FJ [08]

Michael D. Fried and Moshe Jarden, Field arithmetic, third ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 11, Springer-Verlag, Berlin, 2008, Revised by Jarden. MR 2445111

GS [06]

Philippe Gille and Tamás Szamuely, Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, vol. 101, Cambridge University Press, Cambridge, 2006. MR 2266528

Kat [82]

Kazuya Kato, Symmetric bilinear forms, quadratic forms and Milnor $K$ -theory in characteristic two, Invent. Math. 66 (1982), no. 3, 493–510. MR 662605

KM [15]

Daniel Krashen and Eliyahu Matzri, Diophantine and cohomological dimensions, Proc. Amer. Math. Soc. 143 (2015), no. 7, 2779–2788. MR 3336603

KMRT [98]

M.-A. Knus, A. Merkurjev, M Rost, and J.-P. Tignol, The book of involutions, Colloquium Publications, vol. 44, American Mathematical Society, 1998.

Lam [02]

T. Y. Lam, On the linkage of quaternion algebras, Bull. Belg. Math. Soc. Simon Stevin 9 (2002), no. 3, 415–418. MR 2016580

Lan [02]

Serge Lang, Algebra, third ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR 1878556

Mat [16]

Eliyahu Matzri, Symbol length in the Brauer group of a field, Trans. Amer. Math. Soc. 368 (2016), no. 1, 413–427. MR 3413868

MS [82]

A. S. Merkur*′*ev and A. A. Suslin, $K$ -cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 1011–1046, 1135–1136. MR 675529 (84i:12007)

MTW [91]

P. Mammone, J.-P. Tignol, and A. Wadsworth, Fields of characteristic $2$ with prescribed $u$ -invariants, Math. Ann. 290 (1991), no. 1, 109–128. MR 1107665

Ser [94]

Jean-Pierre Serre, Cohomologie galoisienne, fifth ed., Lecture Notes in Mathematics, vol. 5, Springer-Verlag, Berlin, 1994. MR 1324577

Voe [03]

Vladimir Voevodsky, Motivic cohomology with ${\bf Z}/2$ -coefficients, Publ. Math. Inst. Hautes Études Sci. (2003), no. 98, 59–104. MR 2031199

Bibliography24

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1AB [92] Roberto Aravire and Ricardo Baeza, Milnor’s k 𝑘 k -theory and quadratic forms over fields of characteristic two , Comm. Algebra 20 (1992), no. 4, 1087–1107. MR 1154404 (93a:11030)
2AB [10] Jon Kristinn Arason and R. Baeza, La dimension cohomologique des corps de type 𝐂 r subscript 𝐂 r {\bf C}_{\rm r} en caractéristique 𝐩 𝐩 {\bf p} , C. R. Math. Acad. Sci. Paris 348 (2010), no. 3-4, 125–126. MR 2600061
3Alb [68] A. Albert, Structure of algebras , Colloquium Publications, vol. 24, American Math. Soc., 1968.
4AS [78] S. A. Amitsur and D. Saltman, Generic Abelian crossed products and p 𝑝 p -algebras , J. Algebra 51 (1978), no. 1, 76–87. MR 0491789
5Bae [82] R. Baeza, Comparing u 𝑢 u -invariants of fields of characteristic 2 2 2 , Bol. Soc. Brasil. Mat. 13 (1982), no. 1, 105–114. MR 692281
6CGV [17] Adam Chapman, Shira Gilat, and Uzi Vishne, Linkage of quadratic Pfister forms , Comm. Algebra 45 (2017), no. 12, 5212–5226. MR 3691360
7Cha [15] Adam Chapman, Common subfields of p 𝑝 p -algebras of prime degree , Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 4, 683–686. MR 3429179
8Cha [17] , Symbol length of p 𝑝 p -algebras of prime exponent , J. Algebra Appl. 16 (2017), no. 5, 1750136, 9.

TL;DR

Contribution

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Abstract

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Kato-Milne Cohomology and Polynomial Forms

Abstract

keywords:

MSC:

1 Introduction

1.1 The Kato-Milne Cohomology Groups

1.2 CmC_{m}Cm​ and C~p,m\widetilde{C}_{p,m}Cp,m​ Fields

Remark 1.1**.**

1.3 Symbol Length in Hp2(F)H_{p}^{2}(F)Hp2​(F)

1.4 Separable and Inseparable Linkage

2 Fields with bounded upu_{p}up​-invariant

Question 2.2**.**

Example 2.3**.**

Proof.

Lemma 2.4**.**

Proof.

Corollary 2.5**.**

Proof.

3 Symbol Length and ppp-Regular forms

Lemma 3.6**.**

Proof.

Definition 3.7**.**

Lemma 3.8**.**

Proof.

Lemma 3.9**.**

Proof.

Remark 3.10**.**

Lemma 3.11**.**

Proof.

Theorem 3.12** (cf. [8, Theorem 3.2 and Theorem 4.1]).**

Proof.

Corollary 3.13** (cf. [8, Corollary 3.3 and Corollary 4.2]).**

Proof.

Corollary 3.14**.**

Proof.

4 Symbol Length and ppp-Rank

Remark 4.15**.**

Proof.

Proposition 4.16**.**

Proof.

Remark 4.17**.**

Example 4.18**.**

5 Class-Preserving Modifications of Decomposable Differential Forms

Lemma 5.19**.**

Proof.

Proposition 5.20**.**

Remark 5.21**.**

Proof of Proposition 5.2.

Corollary 5.22**.**

Proof.

Corollary 5.23**.**

Proof.

6 Linkage of Decomposable Differential Forms

Theorem 6.24**.**

Proof.

Corollary 6.25**.**

Proof.

Question 6.26**.**

7 Vanishing Cohomology Groups

Theorem 7.27**.**

Proof.

Corollary 7.28**.**

Proof.

Example 7.29**.**

References

1.2 $C_{m}$ and $\widetilde{C}_{p,m}$ Fields

Remark 1.1.

1.3 Symbol Length in $H_{p}^{2}(F)$

2 Fields with bounded $u_{p}$ -invariant

Question 2.2.

Example 2.3.

Lemma 2.4.

Corollary 2.5.

3 Symbol Length and $p$ -Regular forms

Lemma 3.6.

Definition 3.7.

Lemma 3.8.

Lemma 3.9.

Remark 3.10.

Lemma 3.11.

Theorem 3.12 (cf. [8, Theorem 3.2 and Theorem 4.1]).

Corollary 3.13 (cf. [8, Corollary 3.3 and Corollary 4.2]).

Corollary 3.14.

4 Symbol Length and $p$ -Rank

Remark 4.15.

Proposition 4.16.

Remark 4.17.

Example 4.18.

Lemma 5.19.

Proposition 5.20.

Remark 5.21.

Corollary 5.22.

Corollary 5.23.

Theorem 6.24.

Corollary 6.25.

Question 6.26.

Theorem 7.27.

Corollary 7.28.

Example 7.29.