Field of Iterated Laurent Series and its Brauer Group

TL;DR

This paper investigates the symbol length in the Brauer group of iterated Laurent series fields over algebraically closed fields, revealing differences based on characteristic and providing new insights into quadratic forms.

Contribution

It establishes the symbol length as n-1 in characteristic p, contrasting with known results in characteristic not p, and explores properties of Pfister forms over these fields.

Findings

Symbol length in characteristic p is n-1.

Pairs of Pfister forms may lack an (n-1)-fold factor.

Differences in symbol length depend on the field's characteristic.

Abstract

The symbol length of ${_pBr}(k(\!(\alpha_1)\!)\dots(\!(\alpha_n)\!))$ for an algebraically closed field $k$ of $\operatorname{char}(k) \neq p$ is known to be $\lfloor \frac{n}{2} \rfloor$. We prove that the symbol length for the case of $\operatorname{char}(k) = p$ is rather $n-1$. We also show that pairs of anisotropic quadratic or bilinear $n$-fold Pfister forms over this field need not share an $(n-1)$-fold factor.

Equations

$$(\alpha ,\beta {)}_{p,F}=F⟨x,y:x^p=\alpha ,y^p=\beta ,yx=\rho xy⟩$$

$$[\alpha ,\beta {)}_{p,F}=F⟨x,y:x^p-x=\alpha ,y^p=\beta ,yx{y}^{-1}=x+1⟩$$

$$V_{\phi }=\underset{(e_1,\dots ,e_n)\in \{0,1{\}}^{\times n}\setminus \{(0,\dots ,0)\}}{⨁}{F}^2{\alpha }_1^{e_1}\dots {\alpha }_n^{e_n}.$$

$$V_{\psi }=\underset{(e_1,\dots ,e_n)\in \{0,1{\}}^{\times n}\setminus \{(0,\dots ,0)\}}{⨁}{F}^2{\alpha }_1^{e_1}\dots {\alpha }_{n-2}^{e_{n-2}}({\alpha }_{n-1}+1{)}^{e_{n-1}}{\alpha }_{n+1}^{e_{n+1}}.$$

Full text

1 Introduction

Given a prime integer $p$ and a field $F$ of $\operatorname{char}(F)\neq p$ containing a primitive root of unity $\rho$, every cyclic algebra of degree $p$ over $F$ takes the form

[TABLE]

for some $\alpha,\beta\in F^{\times}$. When $\operatorname{char}(F)=p$, every such an algebra takes the form

[TABLE]

for some $\alpha\in F$ and $\beta\in F^{\times}$. These forms are called “(Hilbert) symbol presentations” of the algebras, and the algebras are also called “symbol algebras”. By [7] (when $\operatorname{char}(F)\neq p$ and $\rho\in F$) and [1, Chapter 7, Theorem 30] (when $\operatorname{char}(F)=p$), every class in $\prescript{}{p}{B}r(F)$ is represented by a a tensor product of symbol algebras of degree $p$. The symbol length of a class is the minimal number of symbol algebras required in order to express it, and the symbol length of ${{}_{p}Br}(F)$ is the supremum on the symbol lengths of its classes. Fields $F$ for which $\prescript{}{2}{B}r(F)$ has symbol length 1 are called “linked fields” and are of interest in number theory because of their special arithmetic properties (for example, their $u$-invariant is either 0,1,2,4 or 8 by [6] and [4]).

For complete discretely valued fields, the computation of the symbol length is often easy. In particular, the symbol length of $\prescript{}{p}{B}r(F)$ is $\lfloor\frac{n}{2}\rfloor$ for $F=k(\!(\alpha_{1})\!)\dots(\!(\alpha_{n})\!)$ being the field of iterated Laurent series in $n$ indeterminates over an algebraically closed field $k$ of $\operatorname{char}(k)\neq p$. The goal of this paper is to show that this does not extend to the case of $\operatorname{char}(k)=p$, and in fact the symbol length of $\prescript{}{p}{B}r(F)$ is exactly $n-1$. We also study pairs of quadratic (or bilinear) Pfister forms over such fields when $p=2$ and show that they need not share an $(n-1)$-fold factor.

2 Notation and Terminology

We denote the symbol length of ${{}_{p}Br}(F)$ by $\operatorname{SymL}_{p}(F)$. Given a valued division algebra $D$ (that can be a field, in particular), we denote its residue field by $\overline{D}$ and its value group by $\Gamma_{D}$. We refer the reader to [9] for background on valuation theory on division algebras.

Given a field $F$ of $\operatorname{char}(F)=p>0$, if $[F:F^{p}]<\infty$ then $[F:F^{p}]=p^{r}$ for some nonnegative integer $r$. This $r$ is called the “$p$-rank” of $F$, and denoted by $\operatorname{rank}_{p}(F)$. We say that $I_{q}^{n}F$ is linked if every two quadratic $n$-fold Pfister forms over $F$ share an $(n-1)$-fold Pfister factor. In particular, $F$ is linked when $I_{q}^{2}F$ is linked. We say that $I^{n}F$ is $m$-linked when every $m$ anisotropic bilinear $n$-fold Pfister forms over $F$ share an $(n-1)$-fold factor. When $m=2$ we simply say $I^{n}F$ is linked. When $\operatorname{char}(F)\neq 2$ there is no distinction between $I_{q}^{n}F$ and $I^{n}F$. For background on quadratic and symmetric bilinear forms see [5].

3 The Brauer Group

For the standard case of $\operatorname{char}(k)\neq p$ the following is known:

Proposition 3.1**.**

Let $p$ be a prime integer and $k$ an algebraically closed field of $\operatorname{char}(k)\neq p$. Then $\operatorname{SymL}_{p}(F)=\lfloor\frac{n}{2}\rfloor$ for $F=k(\!(\alpha_{1})\!)\dots(\!(\alpha_{n})\!)$ being the field of iterated Laurent series in $n$ indeterminates.

Proof.

It follows from [9, Theorem 7.80]. ∎

We recall two theorems from the literature that are useful for our cause:

Theorem 3.2** ([8, Theorem 1]).**

Suppose $F$ is a Henselian valued field, $D$ and $E$ are division algebras over $F$ such that

$D$* is defectless,* 2. 2.

$\overline{D}\otimes\overline{E}$* is a division algebra, and* 3. 3.

$\Gamma_{D}\cap\Gamma_{E}=\Gamma_{F}$.

Then $D\otimes E$ is a division algebra.

Theorem 3.3** ([2, Theorem 3.3]).**

Let $F$ be a maximally complete field of $\operatorname{char}(F)=p>0$ with ${\operatorname{dim}}_{\mathbb{F}_{p}}(\Gamma_{F}/p\Gamma_{F})=n$. Write ${\operatorname{dim}}_{\mathbb{F}_{p}}(\overline{F}/\wp(\overline{F}))=m$. Then the symbol length of ${{}_{p}Br}(F)$ is at most $n-1$ when $m<n$ and $n$ when $m\geqslant n$.

We are now ready to prove the main results:

Lemma 3.4**.**

Let $p$ be a prime integer and $k$ a field of $\operatorname{char}(k)=p$. Then $[\alpha_{2}^{-1},\alpha_{1})_{p,F}\otimes[\alpha_{3}^{-1},\alpha_{2})_{p,F}\otimes\dots\otimes[\alpha_{n}^{-1},\alpha_{n-1})_{p,F}$ is a division algebra over $F=k(\!(\alpha_{1})\!)\dots(\!(\alpha_{n})\!)$.

Proof.

We prove this by induction. For $n=2$, the algebra $[\alpha_{2}^{-1},\alpha_{1})_{p,F}$ is indeed a division algebra: The étale extension $K=F[x:x^{p}-x=\alpha_{2}^{-1}]$ is a field extension because the value of $\alpha_{2}^{-1}$ with respect to the right-to-left $(\alpha_{1},\alpha_{2})$-adic valuation is $(0,-1)$, and so if a solution to the equation $z^{p}-z=\alpha_{2}^{-1}$ existed in $F$, the root $z$ would be of value $(0,-\frac{1}{p})$, but $\Gamma_{F}=\mathbb{Z}\times\mathbb{Z}$, contradiction. Now, the valuation extends to the field $K$ and $\Gamma_{K}=\mathbb{Z}\times\frac{1}{p}\mathbb{Z}$. If $\alpha_{1}$ were a norm in $K/F$, since its value is $(1,0)$, there would exist an element in $K$ of value $(\frac{1}{p},0)$, but there is no such element. Hence $[\alpha_{2}^{-1},\alpha_{1})_{p,F}$ is a division algebra.

Assume that $[\alpha_{2}^{-1},\alpha_{1})_{p,F}\otimes[\alpha_{3}^{-1},\alpha_{2})_{p,F}\otimes\dots\otimes[\alpha_{m}^{-1},\alpha_{m-1})_{p,F}$ is a division algebra for every $m\in\{2,\dots,n-1\}$ and any field $F$ of iterated Laurent series in $m$ indeterminates over any field $k$ of $\operatorname{char}(k)=p$. Let us now look at $[\alpha_{2}^{-1},\alpha_{1})_{p,F}\otimes[\alpha_{3}^{-1},\alpha_{2})_{p,F}\otimes\dots\otimes[\alpha_{n}^{-1},\alpha_{n-1})_{p,F}$ over $F=k(\!(\alpha_{1})\!)\dots(\!(\alpha_{n})\!)$. Write $F=K(\!(\alpha_{n})\!)$ where $K=k(\!(\alpha_{1})\!)\dots(\!(\alpha_{n-1})\!)$, $D=[\alpha_{2}^{-1},\alpha_{1})_{p,F}\otimes[\alpha_{3}^{-1},\alpha_{2})_{p,F}\otimes\dots\otimes[\alpha_{n-1}^{-1},\alpha_{n-2})_{p,F}$ and $E=[\alpha_{n}^{-1},\alpha_{n-1})_{p,F}$. Consider the $\alpha_{n}$-adic valuation on $F$. Then $\overline{E}=K[q:q^{p}=\alpha_{n-1}]=k(\!(\alpha_{1})\!)\dots(\!(\alpha_{n-2})\!)(\!(q)\!)$. In addition, the algebra $D$ is defectless with respect to this valuation, and we have $\overline{D}\otimes\overline{E}=[\alpha_{2}^{-1},\alpha_{1})_{p,\overline{E}}\otimes[\alpha_{3}^{-1},\alpha_{2})_{p,\overline{E}}\otimes\dots\otimes[\alpha_{n-2}^{-1},\alpha_{n-3})_{p,\overline{E}}\otimes[q^{-1},\alpha_{n-2})_{p,\overline{E}}$, which is a division algebra by the induction hypothesis. The conditions of Theorem 3.2 are therefore satisfied, and so $D\otimes E$ is a division algebra over $F$, and the induction is complete. ∎

Theorem 3.5**.**

Let $p$ be a prime integer, $k$ a perfect field of $\operatorname{char}(k)=p$ and $F=k(\!(\alpha_{1})\!)\dots(\!(\alpha_{n})\!)$. Then $\operatorname{SymL}_{p}(F)=n-1$ when ${\operatorname{dim}}_{\mathbb{F}_{p}}(k/\wp(k))<n$ (e.g., when $k$ is algebraically closed), and $\operatorname{SymL}_{p}(F)=n$ when ${\operatorname{dim}}_{\mathbb{F}_{p}}(k/\wp(k))\geqslant n$.

Proof.

Suppose ${\operatorname{dim}}_{\mathbb{F}_{p}}(k/\wp(k))<n$. Consider the right-to-left $(\alpha_{1},\dots,\alpha_{n})$-adic valuation on $F$. Since the residue field is algebraically closed, by Theorem 3.3 the symbol length of ${{}_{p}Br}(F)$ is at most $n-1$. On the other hand, by Lemma 3.4 there exists a class of symbol length $n-1$ in ${{}_{p}Br}(F)$, so the symbol length of ${{}_{p}Br}(F)$ is exactly $n-1$.

Now suppose ${\operatorname{dim}}_{\mathbb{F}_{p}}(k/\wp(k))\geqslant n$. By Theorem 3.3 the symbol length of ${{}_{p}Br}(F)$ is at most $n$. Take $\beta_{1},\dots,\beta_{n}\in k$ to be $\mathbb{F}_{p}$-linearly independent elements in $k/\wp(k)$, and consider the algebra $D=[\beta_{1},\alpha_{1})_{p,F}\otimes\dots\otimes[\beta_{n},\alpha_{n})_{p,F}$. This algebra is generated over $F$ by $x_{1},y_{1},\dots,x_{n},y_{n}$ satisfying $x_{i}^{p}-x_{i}=\beta_{i}$, $y_{i}^{p}=\alpha_{i}$ and $y_{i}x_{i}y_{i}^{-1}=x_{i}+1$, $y_{i}y_{j}=y_{j}y_{i}$ and $x_{i}x_{j}=x_{j}x_{i}$ for any $i,j\in\{1,\dots,n\}$ with $i\neq j$. Hence, this algebra can also be viewed as the skew field of twisted iterated Laurent series $L(\!(y_{1};\sigma_{1})\!)\dots(\!(y_{n};\sigma_{n})\!)$ over $L=k[x_{1},\dots,x_{n}:x_{i}^{p}-x_{i}=\beta_{i}\forall i\in\{1,\dots,n\}]$ where each $\sigma_{i}$ is the automorphism of $L$ mapping $x_{i}$ to $x_{i}+1$ and every other $x_{j}$ to itself (see [9, Section 1.1.3]). In particular, $D$ is a division algebra whose symbol length is $n$, and so $\operatorname{SymL}_{p}(F)=n$. ∎

Corollary 3.6**.**

When $\operatorname{char}(k)=2$, the field $F=k(\!(\alpha)\!)(\!(\beta)\!)(\!(\gamma)\!)$ is not linked.

Note that this means that $I_{q}^{2}F$ is not linked for $F=k(\!(\alpha)\!)(\!(\beta)\!)(\!(\gamma)\!)$. In the next section we generalize this fact to arbitrary quadratic and bilinear Pfister forms.

4 Quadratic and Bilinear Forms

We first cover the standard case:

Proposition 4.7**.**

Given an algebraically closed field $k$ of $\operatorname{char}(k)\neq 2$, $I^{n}F$ is linked for $F=k(\!(\alpha_{1})\!)\dots(\!(\alpha_{n+1})\!)$.

Proof.

Consider the $n$-fold Pfister forms $\varphi=\langle\!\langle a_{1},\dots,a_{n}\rangle\!\rangle$ and $\psi=\langle\!\langle b_{1},\dots,b_{n}\rangle\!\rangle$. By Hensel’s Lemma, the $a$-s and $b$-s can be assumed to be products of powers of the $\alpha$-s. Consider the group $F^{\times}/(F^{\times})^{2}$ as a vector space $V\cong\mathbb{F}_{2}^{\times(n+1)}$. By the identity $\langle\!\langle a,b\rangle\!\rangle=\langle\!\langle ab,b\rangle\!\rangle$, and the assumption that $\varphi$ and $\psi$ are anisotropic, the elements $a_{1},\dots,a_{n}$ span an $n$-dimensional subspace $W_{a}$ of $V$ and the elements $b_{1},\dots,b_{n}$ span an $n$-dimensional subspace $W_{b}$ of $V$. Therefore, $W_{a}\cap W_{b}$ is at least of dimension $n-1$. Take a basis $c_{1},\dots,c_{n-1}$ for this $(n-1)$-dimensional subspace of the intersection, and $\langle\!\langle c_{1},\dots,c_{n-1}\rangle\!\rangle$ is a common factor of $\varphi$ and $\psi$. ∎

When $\operatorname{char}(k)=2$ we need to distinguish between quadratic and bilinear forms.

Theorem 4.8**.**

Given a field $k$ of $\operatorname{char}(k)=2$, $I_{q}^{n}F$ is not linked for $F=k(\!(\alpha_{1})\!)\dots(\!(\alpha_{n+1})\!)$.

Proof.

Let $\varphi=\langle\!\langle\alpha_{1},\dots,\alpha_{n-2},\alpha_{n-1},\alpha_{n}^{-1}]\!]$ and $\psi=\langle\!\langle\alpha_{1},\dots,\alpha_{n-2},\alpha_{n},\alpha_{n+1}^{-1}]\!]$. The form $\varphi\perp\psi$ is Witt equivalent to $\omega=\langle\!\langle\alpha_{1},\dots,\alpha_{n-2}\rangle\!\rangle\otimes[1,\alpha_{n}^{-1}+\alpha_{n+1}^{-1}]\perp\alpha_{n-1}\langle\!\langle\alpha_{1},\dots,\alpha_{n-2}\rangle\!\rangle\otimes[1,\alpha_{n}^{-1}]\perp\alpha_{n}\langle\!\langle\alpha_{1},\dots,\alpha_{n-2}\rangle\!\rangle\otimes[1,\alpha_{n+1}^{-1}]$. For $\varphi$ and $\psi$ to share a common $(n-1)$-fold factor, $\omega$ must be isotropic, but it is not. This can be seen by considering the right-to-left $(\alpha_{1},\dots,\alpha_{n+1})$-adic valuation. The values modulo $(2\mathbb{Z})^{\times(n+1)}$ of all the symplectic blocks in $\omega$ are distinct, and so the form must be anisotropic. ∎

Theorem 4.9**.**

Let $F$ be a field of $\operatorname{char}(F)=2$ and $\operatorname{rank}_{2}(F)\geqslant n+1$. Then $I^{n}F$ is not linked.

Proof.

Let $\langle\!\langle\alpha_{1},\dots,\alpha_{n+1}\rangle\!\rangle$ be an anisotropic bilinear $n$-fold Pfister form over $F$. Consider the forms $\varphi=\langle\!\langle\alpha_{1},\dots,\alpha_{n-2},\alpha_{n-1},\alpha_{n}\rangle\!\rangle$ and $\psi=\langle\!\langle\alpha_{1},\dots,\alpha_{n-2},\alpha_{n-1}+1,\alpha_{n+1}\rangle\!\rangle$. Recall that an anisotropic bilinear Pfister form $\tau$ has a “pure subform” $\tau^{\prime}$ which is the unique anisotropic symmetric bilinear form satisfying $\tau^{\prime}\perp\langle 1\rangle=\tau$ (see [5, Chapter 1, Section 6]). We therefore denote by $\varphi^{\prime}$ and $\psi^{\prime}$ the pure subforms of $\varphi$ and $\psi$, respectively. The set $D(\varphi^{\prime})\cup\{0\}$ is the $(2^{n}-1)$-dimensional $F^{2}$-vector space

[TABLE]

The set $D(\psi^{\prime})\cup\{0\}$ is the $(2^{n}-1)$-dimensional $F^{2}$-vector space

[TABLE]

In order for $\psi$ and $\varphi$ to share an $(n-1)$-fold factor, the intersection $V_{\varphi}\cap V_{\psi}$ should be of dimension at least $2^{n-1}-1$. However, the intersection of these two spaces is $W\oplus\alpha_{n-1}W$ where $W=\bigoplus_{(e_{1},\dots,e_{n-2})\in\{0,1\}^{\times(n-1)}\setminus\{(0,\dots,0)\}}F^{2}\alpha_{1}^{e_{1}}\dots\alpha_{n-2}^{e_{n-2}}$, which is of dimension $(2^{n-2}-1)\cdot 2=2^{n-1}-2$, i.e., less than $2^{n-1}-1$. Therefore, the forms do not share an $(n-1)$-fold factor. ∎

Corollary 4.10**.**

Given a field $F$ of $\operatorname{char}(F)=2$ with $I^{n}F\neq 0$, the following are equivalent:

$I^{n}F$* is linked.* 2. 2.

$I^{n}F$* is 3-linked.* 3. 3.

The 2-rank of $F$ is $n$.

Proof.

By the assumption, the 2-rank of $F$ is at least $n$. By Theorem 4.9, (1) implies (3). By [3, Corollary 3.2], (3) implies (2), and clearly (2) implies (1). ∎

Note that when $k$ is an algebraically closed field of $\operatorname{char}(k)=2$, the field $F=k(\!(\alpha_{1})\!)\dots(\!(\alpha_{n+1})\!)$ satisfies the conditions of Theorems 4.8 and 4.9.

Acknowledgements

The author thanks Jean-Pierre Tignol for the helpful discussions, and the anonymous referee for the constructive suggestions.