Generic polynomials for cyclic function field extensions over certain finite fields

TL;DR

This paper determines all generic polynomials for specific cyclic function field extensions over finite fields where the field size satisfies certain congruence conditions, expanding understanding of such algebraic structures.

Contribution

It explicitly finds all generic polynomials for geometric yclic function field extensions over finite fields with particular size and characteristic constraints.

Findings

Identifies all generic polynomials for the specified cyclic extensions.

Provides explicit constructions under given finite field conditions.

Enhances the classification of function field extensions with cyclic Galois groups.

Abstract

In this paper, we find all the generic polynomials for geometric $\ell$-cyclic function field extensions over the finite fields $\mathbb{F}_q$ where $q= p^n$, $p$ prime integer such that $q \equiv -1 \mod \ell$ and $(\ell , p)=1$.

Equations

$$P_{u,\alpha }^{\ell }(X)=X^{\ell }-\ell u{X}^{\ell -2}+\sum _{s=1}^{\iota }{c}_{s,\ell }{u}^s{X}^{\ell -2s}-\alpha$$

$$c_{s,j}^r=-\sum _{k=1}^s\left(\genfrac{}{}{0ex}{}{2j+r}{k}\right)c_{s-k,j-1-k}\in {\mathbb{F}}_p$$

$$Q_u^l(X)=X^l-\ell u{X}^{l-2}+\sum _{s=1}^{\iota }{c}_{s,l}{u}^s{X}^{l-2s}$$

$$P_{u,\alpha }^{\ell }(X)=P_{l_t,u^{l/l_t},\alpha }(\cdots (Q_{l_2,u^{l_1}}(Q_{l_1,u}(X))$$

$$\begin{array}[]{ccl}y^{\ell}&=&(w+\sigma(w))^{\ell}=\sum_{i=0}^{\ell}\binom{\ell}{i}w^{i}\sigma(w)^{\ell-i}\\ &=&w^{\ell}+\sigma(w)^{\ell}+\sum_{i=1}^{\ell-1}\binom{\ell}{i}w^{i}\sigma(w)^{\ell-i}\\ &=&a+\sigma(a)+\sum_{i=1}^{\iota-(1-r)}\binom{\ell}{i}w^{i}\sigma(w)^{i}(\sigma(w)^{\ell-2i}+w^{\ell-2i})+(1-r)\binom{\ell}{\iota}w^{\iota}\sigma(w)^{\iota}\\ &=&a+\sigma(a)+\sum_{i=1}^{\iota-(1-r)}\binom{\ell}{i}u^{i}(\sigma(w)^{\ell-2i}+w^{\ell-2i})+(1-r)\binom{\ell}{\iota}u^{\iota}\\ \end{array}$$

$${\omega }_j^r=w^{2j+r}+\sigma (w{)}^{2j+r}=y^{2j+r}-\sum _{k=1}^j(\genfrac{}{}{0ex}{}{2j+r}{k})u^k{\omega }_{j-k}^r$$

$${\omega }_0^1=y,$$

$${\omega }_1^1=y^3-3uy,$$

$${\omega }_2^1=y^5-5u{y}^3+5u^{2}y,$$

$${\omega }_1^0=y^2-2u$$

$${\omega }_2^0=y^4-4u{y}^2+2u^2$$

$${\omega }_j^r=\sum _{s=0}^j{c}_{s,j}^r{u}^s{y}^{2j+r-2s}$$

$$\begin{array}[]{ccl}\omega_{j+1}&=&y^{2(j+1)+r}-\sum_{k=1}^{j+1}\binom{2(j+1)+r}{k}u^{k}\omega_{j+1-k}^{r}\\ &=&y^{2(j+1)+r}-\sum_{k=1}^{j+1}u^{k}\binom{2(j+1)+r}{k}\sum_{s=0}^{j+1-k}c_{s,j+1-k}^{r}u^{s}y^{2(j+1)+r-2(s+k)}\\ &=&y^{2(j+1)+r}-\sum_{k=1}^{j+1}\sum_{s^{\prime}=k}^{j+1}\binom{2(j+1)+r}{k}u^{s^{\prime}}c_{s^{\prime}-k,j+1-k}^{r}y^{2(j+1)+r-2s^{\prime}}\ \text{ where \ $s^{\prime}:=s+k$}\\ &=&y^{2(j+1)+r}+\sum_{s^{\prime}=1}^{j+1}(-\sum_{k=1}^{s^{\prime}}\binom{2(j+1)+r}{k}c_{s^{\prime}-k,j+1-k}^{r})u^{s^{\prime}}y^{2(j+1)+r-2s^{\prime}}\end{array}$$

$$\begin{array}[]{ccl}0&=&(1-1)^{2n}\\ &=&\sum_{k=0}^{2n}\binom{2n}{k}(-1)^{k}\\ &=&\sum_{k=0}^{n}\binom{2n}{k}(-1)^{k}+\sum_{k=n+1}^{2n}\binom{2n}{k}(-1)^{k}\\ &=&\sum_{k=0}^{n}\binom{2n}{k}(-1)^{k}+\sum_{k=0}^{n-1}\binom{2n}{k}(-1)^{k}\\ &=&2\sum_{k=0}^{n-1}\binom{2n}{k}(-1)^{k}+\binom{2n}{n}(-1)^{n}\end{array}$$

$$\begin{array}[]{ccl}c_{j+1,j+1}^{0}&=&-\sum_{k=1}^{j+1}\binom{2(j+1)}{k}c_{j+1-k,j+1-k}^{0}\\ &=&-2(-1)^{j+1}\sum_{k=1}^{j}\binom{2(j+1)}{k}(-1)^{k}-\binom{2(j+1)}{j+1}\\ &=&2(-1)^{j+1}-2(-1)^{j+1}\sum_{k=0}^{j}\binom{2(j+1)}{k}(-1)^{k}-\binom{2(j+1)}{j+1}\\ &=&(-1)^{j+1}2\text{ using $(1)$}\end{array}$$

$$\begin{array}[]{ccl}&&c_{j+1,j+1}^{1}\\ &=&-\sum_{k=1}^{j+1}\binom{2(j+1)+1}{k}c_{j+1-k,j+1-k}^{0}\\ &=&-(-1)^{j+1}\sum_{k=1}^{j+1}\binom{2(j+1)+1}{k}(-1)^{k}(2(j+1-k)+1)\\ &=&-(-1)^{j+1}\sum_{k=1}^{j+1}\binom{2(j+1)+1}{k}(-1)^{k}(2(j+1)+1)+2(-1)^{j+1}\sum_{k=1}^{j}\binom{2(j+1)+1}{k}k(-1)^{k}\\ &=&-(-1)^{j+1}(2(j+1)+1)(\sum_{k=1}^{j+1}(\binom{2(j+1)+1}{k}-2\binom{2(j+1)}{k-1})(-1)^{k})\\ &=&-(-1)^{j+1}(2(j+1)+1)(\sum_{k=1}^{j}\binom{2(j+1)}{k}(-1)^{k}-\sum_{k=1}^{j+1}\binom{2(j+1)}{k-1})(-1)^{k})\\ &=&-(-1)^{j+1}(2(j+1)+1)(\sum_{k=1}^{j+1}\binom{2(j+1)}{k}(-1)^{k}+\sum_{k=0}^{j}\binom{2(j+1)}{k})(-1)^{k})\\ &=&(-1)^{j+1}(2(j+1)+1)\text{ using (1)}\end{array}$$

$$P_{u,{\alpha }_1}^{l_1}(X)$$

$$P_{u^{l_1},{\alpha }_2}^{l_2}(Q_{l_1,u}(X))$$

$$P_{u^{l/l_t},\alpha }^{l_t}(\cdots (Q_{u^{l_1}}^{l_2}(Q_u^{l_1}(X))$$

$$P_{u,\alpha }^{\ell }(X)=P_{l_t,u^{l/l_t},\alpha }(\cdots (Q_{l_2,u^{l_1}}(Q_{l_1,u}(X))$$

$$z^{{\ell }_1}=w_{1,1},\cdots ,w_{1,f_1}^{{\ell }_2}=w_{2,1},\cdots ,w_{s,f_s}^{{\ell }_s}=a$$

$$L(\xi )/K(\xi )(w_{1,1})/\cdots \cdots /K(\xi )(w_{s,f_s})/K(\xi ).$$

$$(z\sigma (z){)}^{{\ell }_1}=w_{1,1}\sigma (w_{1,1}),\cdots ,(w_{1,f_1}\sigma (w_{1,f_1}){)}^{{\ell }_2}=w_{2,1}\sigma (w_{2,1}),\cdots ,(w_{s,f_s}\sigma (w_{s,f_s}){)}^{{\ell }_s}=a\sigma (a)$$

$$[K(z\sigma (z))/K]=[K(w_{1,1}\sigma (w_{1,1}))/K]=\cdots =[K(w_{s,f_s}\sigma (w_{s,f_s}))/K]=1$$

$$[K(z\sigma (z))/K]=[K(w_{s,1}\sigma (w_{s,1}))/K]$$

$$(w_{s,j}\sigma (w_{s,j}){)}^4=w_{s,j+2}\sigma (w_{s,j+2})$$

$$[K(z\sigma (z))/K]=[K(w_{s,1}\sigma (w_{s,1}))/K]=[K(w_{s,2}\sigma (w_{s,2}))/K]=[K(w_{s,e_s}\sigma (w_{s,e_s}))/K]$$

$$(w_{s,e_s}\sigma (w_{s,e_s}){)}^2=a\sigma (a)$$

$$\zeta z+\sigma (\zeta z)=\frac{\xi -{\xi }^{-1}}{(\xi +{\xi }^{-1}{)}^2-4}(z_1+\xi z_2)+\frac{{\xi }^{-1}-\xi }{(\xi +{\xi }^{-1}{)}^2-4}(z_1+{\xi }^{-1}{z}_2)=z_2.$$

$${\zeta }^{\prime }z+\sigma ({\zeta }^{\prime }z)=\frac{{\xi }^{-2}-1}{(\xi +{\xi }^{-1}{)}^2-4}(z_1+\xi z_2)+\frac{{\xi }^2-1}{(\xi +{\xi }^{-1}{)}^2-4}(z_1+{\xi }^{-1}{z}_2)=z_1.$$

Full text

Throughout the paper we will adopt the following notations unless mentioned differently. $K$ denotes a one variable function field with field of constants $\mathbb{F}_{q}$ where $q=p^{n}$, $p$ prime integer. Given $x\in K$, we denote $\mathcal{O}_{K,x}$ is the integral closure of $\mathbb{F}_{q}[x]$ in $L$. Let $L/K$ be a cyclic Galois extension of degree $\ell$ with $(\ell,p)=1$. We write $\ell=\prod_{i=1}^{s}\ell_{i}^{f_{i}}$ where $\ell_{i}$ are distinct prime integers and $f_{i}$ positive integer. Let $\xi$ be a primitive $\ell^{th}$-root of unity. We suppose that $q\equiv-1\mod\ell$. So that, by [2, Theorem 2.10], $\mathbb{F}_{q}$ do not contains any primitive $\ell^{th}$ root of unity. More precisely, we have $[\mathbb{F}_{q}(\xi):\mathbb{F}_{q}]=2$ and the minimal polynomial of $\xi$ over $\mathbb{F}_{q}$ is $X^{2}-(\xi+\xi^{-1})X+1$ and $(\xi+\xi^{-1})\in\mathbb{F}_{q}$. We denote by $\sigma$ the generator of $Gal(\mathbb{F}_{q}(\xi),\mathbb{F}_{q})$, we have that $\sigma(\xi)=\xi^{-1}$.

In this paper, our main result finds all the generic polynomials for geometric cyclic function field extensions over those finite fields $\mathbb{F}_{q}$ (Theorem 1.5). We find a one parameter family of generic polynomials of a cyclic extensions when $\ell$ is odd and a two parameters family of generic polynomials when $\ell$ is even (Corollary 1.8). We also classify cyclic extensions up to isomorphism over those finite field $\mathbb{F}_{q}$ (Lemma 1.10). Note that, in particular, this permits to classify all the geometric cyclic extensions of degree $3$, $4$ and $6$ over any finite fields $\mathbb{F}_{q}$. We describe the Galois action on a generator with a minimal polynomial in our form (Corollary 1.11). We end the paper with the study of the ramification in term of our generation (Theorem 2.1). In particular, we find that under our assumptions the ramified places are of even degree.

1 Generic polynomials for cyclic extensions

Lemma 1.1**.**

Let $L/K$ be a cyclic extension of degree $\ell$ with $q\equiv-1\ mod\ \ell$. Suppose $w$ is a Kummer generator for $L(\xi)/K(\xi)$ whose minimal polynomial is of the form $X^{\ell}-a$, $y:=\sigma(w)+w$ is a generator for $L/K$ and $u:=\sigma(w)w\in K$. We write $\ell=2\iota+r$, where $r=0,1$. Then, the minimal polynomial of $y$ over $K$ is of the form

[TABLE]

where $\alpha=a+\sigma(a)$ and $u^{\ell}=a\sigma(a)$, the coefficients $c^{r}_{s,j}\in\mathbb{F}_{p}$ and are defined recursively as

[TABLE]

*for $0\leq s\leq j$ and $1\leq j\leq\ell$ and $c_{0,j}^{r}=1$ for all $j\geq 0$. In particular, we have that $c_{1,j}^{r}=-(2j+r)$, $c_{j,j}^{1}=(-1)^{j}(2j+1)$ and $c_{j,j}^{0}=(-1)^{j}2$.

Defining,*

[TABLE]

for any $l$ integer where the $c_{s,l}$ are define recursively as before, we have that if $\ell=\prod_{i=1}^{t}l_{i}$ where $l_{i}$ are not necessarily distinct factors of $\ell$, then

[TABLE]

Proof.

Under the assumption of the theorem we determine the minimal polynomial of $y=w+\sigma(w)$. For if,

[TABLE]

Note that $\ell-2i$ is the same parity of $\ell$.

We set $\omega_{j}^{r}=w^{2j+r}+\sigma(w)^{2j+r}$ for $0\leq j\leq\frac{\ell-1}{2}$ and $(j,r)\neq 0$ and $\omega_{0}^{0}=1$, then

[TABLE]

We have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

By induction in $j$ for $r=0,1$, we prove that

[TABLE]

where the coefficients are obtained recursively as $c_{s,j}^{r}=-\sum_{k=1}^{s}\binom{2j+r}{k}c_{s-k,j-k}^{r}$, $1\leq s\leq j$ and $j\geq 1$ and $c_{0,j}^{r}=1$ for all $j\geq 0$. We also prove that $c_{1,j}^{r}=-(2j+r)$, $c_{j,j}^{1}=(-1)^{j}(2j+1)$ and $c_{j,j}^{0}=(-1)^{j}2$.

From the computation above we see that this property is true for $j=1$. Suppose that this property is true for $\omega_{t}^{r}$, $1\leq t\leq j$, for some fixed $j$. We want to prove it remain true for $w_{j+1}^{r}$. Using the induction assumption, we obtain

[TABLE]

Thus, $c_{s,j+1}^{r}=-\sum_{k=1}^{s}\binom{2(j+1)+r}{k}c_{s-k,j+1-k}^{r}$, for $1\leq s\leq j+1$.

Clearly, $c_{1,j+1}^{r}=-(2(j+1)+r)$.

Note that

[TABLE]

Using the induction assumption, we have,

[TABLE]

and

[TABLE]

Moreover, note that when $\ell=\prod_{i=1}^{t}l_{i}$ where $l_{i}$ are non necessarily distinct factor of $\ell$, the minimal polynomial of $y$ over $L^{l_{1}}$ the fixed field of $L$ by the subgroup of $Gal(L/K)$ isomorphic to $\mathbb{Z}/l_{1}\mathbb{Z}$ is

[TABLE]

where $\alpha_{1}=w^{l_{1}}+\sigma(w^{l_{1}})$.

We let $Q^{l_{1}}_{u}=P^{l_{1}}_{u,\alpha_{1}}+\alpha_{1}$. Since $y$ is a generator for $L/K$, we have that $[K(w^{l_{1}}+\sigma(w^{l_{1}})):K]=\ell/l_{1}$ and $w^{l_{1}}$ is a Kummer generator for $[K(\xi)(w^{l_{1}}):K(\xi)]$, thus the minimal of $y$ over $L^{l_{1}l_{2}}$ the fixed field of $L$ by the subgroup of $Gal(L/K)$ isomorphic to $\mathbb{Z}/l_{1}l_{2}\mathbb{Z}$ is

[TABLE]

where $\alpha_{2}=w^{l_{1}l_{2}}+\sigma(w^{l_{1}l_{2}})$. Recursively, we obtain that the minimal polynomial $y$ of $L/K$ is

[TABLE]

where $Q^{l_{k}}_{u}=P^{l_{k}}_{u,\alpha_{k}}+\alpha_{k}$ and $\alpha_{k}=w^{l_{1}l_{2}\cdots l_{k}}+\sigma(w^{l_{1}l_{2}\cdots l_{k}})$.

By uniqueness of the minimal polynomial of $y$ over $K$, we obtain that

[TABLE]

∎

Remark 1.2**.**

The previous lemma hold for general field extensions over a field of positive characteristic. 2. 2.

Note that when $2|\ell$, the polynomial $P^{\ell}_{u,\alpha}(X)$ only involve even degree monomials and when $(\ell,2)=1$, the polynomial $P^{\ell}_{u,\alpha}(X)$ only involve odd degree monomials. 3. 3.

We list some of those polynomials

• $\cdot$

$P^{3}_{u,\alpha}(X)=X^{3}-3uX-\alpha$,

• $\cdot$

$P^{5}_{u,\alpha}(X)=X^{5}-5uX^{3}+5u^{2}X-\alpha$,

• $\cdot$

$P^{7}_{u,\alpha}(X)=X^{7}-7uX^{5}+14u^{2}X^{3}-7u^{3}X-\alpha$,

• $\cdot$

$P^{9}_{u,\alpha}(X)=X^{9}-9uX^{7}+27u^{2}X^{5}-30u^{3}X^{3}+9u^{4}X-\alpha$,

• $\cdot$

$P^{11}_{u,\alpha}(X)=X^{11}-11uX^{9}+44u^{2}X^{7}-77u^{3}X^{5}+55u^{4}X^{3}-11u^{5}X-\alpha$**

• $\cdot$

$P^{13}_{u,\alpha}(X)=X^{13}-13uX^{11}+65u^{2}X^{9}-156u^{3}X^{7}+182u^{4}X^{5}-91u^{5}X^{3}+13u^{6}X-\alpha$**

• $\cdot$

$P^{2}_{u,\alpha}(X)=X^{2}-2u-\alpha$,

• $\cdot$

$P^{4}_{u,\alpha}(X)=X^{4}-4uX^{2}+2u^{2}-\alpha$**

• $\cdot$

$P^{6}_{u,\alpha}(X)=X^{6}-6uX^{4}+9u^{2}X^{2}-2u^{3}-\alpha$**

• $\cdot$

$P^{8}_{u,\alpha}(X)=X^{8}-8uX^{6}+20u^{2}X^{4}-16u^{3}X^{2}+2u^{4}-\alpha$.

We recognize $P^{3}_{u,\alpha}(X)$ when $u=1$ being the generic polynomials obtained in **[3]** and **[4]**. We will prove that those polynomials $P^{\ell}_{u,\alpha}(X)$ are generic polynomials for cyclic extensions of degree $\ell$ over $\mathbb{F}_{q}$ when $q\equiv-1\ mod\ 3$.

Lemma 1.3**.**

Let $L/K$ be a geometric cyclic extension of degree $\ell$ with $q\equiv-1\ mod\ \ell$. For any $z$ Kummer generator for $L(\xi)/K(\xi)$, $z\sigma(z)\in K$.

Proof.

By [1, Theorem 5.8.5], we know that $L(\xi)/K(\xi)$ is a Kummer extension thus there exists a Kummer generator $z$ whose minimal polynomial is $X^{\ell}-a$ with $a\in K(\xi)$. Note that $\sigma(z)$ is also a Kummer generator for $L(\xi)/K(\xi)$ with minimal polynomial $X^{\ell}-\sigma(a)$.

We can write $L(\xi)/K(\xi)$ as a tower of prime degree extension

[TABLE]

and we get a tower of cyclic prime degree extension

[TABLE]

We set $w_{0,1}=z$, $w_{s+1,f_{s+1}}=a$ and $f_{0}=f_{s+1}=1$.

We want to prove that $z\sigma(z)\in K$. Since $\sigma(z\sigma(z))=z\sigma(z)$, $z\sigma(z)\in L$ and we have

[TABLE]

Similarly, $w_{i,j}\sigma(w_{i,t_{i}})\in K(w_{i,t_{i}})$, for any $1\leq i\leq s$ and $1\leq t_{i}\leq f_{i}$.

We have two cases, either all the $\ell_{i}$ are odd or one of the $\ell_{i}$ is equal to $2$, in which case we can suppose without loss of generality that $\ell_{s}=2$.

Case 1: all $\ell_{j}$ are odd, for $1\leq j\leq s$.

Since $(w_{i,j}\sigma(w_{i,j}))^{\ell_{k}}=w_{k,l}\sigma(w_{k,l})$, for any $0\leq i\leq s$ and $1\leq t_{i}\leq f_{i}$ with

• $\cdot$

$k=i+1$ and $l=1$, when $k=i$ and $j=f_{i}$,

• $\cdot$

$l=j+1$ and $i=k$, if $0\leq j<f_{i}$,

then either $w_{i,j}\sigma(w_{i,j})\in K(w_{k,l}\sigma(w_{k,l}))$ or $K(w_{i,j}\sigma(w_{i,j}))/K(w_{k,l}\sigma(w_{k,l}))$ is a extension of degree $\ell_{k}$ in $L/K$. But the later case is impossible since $K(w_{i,j}\sigma(w_{i,j}))/K(w_{k,l}\sigma(w_{k,l}))$ is a subextension in the cyclic extension $L/K$ so it would be a Kummer extension, which is contradictory to the assumption that $q\equiv-1\ mod\ \ell$ implying that $q\equiv-1\ mod\ \ell_{k}$ (see [1, Theorem 5.8.5]). From this, we can prove that

[TABLE]

And thus, $z\sigma(z)\in K$. 2. 2.

Case 2: one of the $\ell_{i}$ is equal to $2$, in which case we can suppose without loss of generality that $\ell_{s}=2$.

From Case 1, we know that

[TABLE]

1. (a)

If $f_{s}=1$, we have $(w_{s,1}\sigma(w_{s,1}))^{2}=a\sigma(a)$.

Thus either $w_{s,j}\sigma(w_{s,j})\in K(w_{s,j+1}\sigma(w_{s,j+1}))$ or $w_{s,j}\sigma(w_{s,j})$ is a Kummer generator for $K(w_{s,1}\sigma(w_{s,1}))/K$ and thus for $K(\xi)(w_{s,1}\sigma(w_{s,1}))/K(\xi)$. But then $w_{s,1}\sigma(w_{s,1})$ and $w_{s,1}$ are two Kummer generators for $K(\xi)(w_{s,1}\sigma(w_{s,1}))/K(\xi)$. Hence, by [1, Theorem 5.8.5], there is $d\in K(\xi)$ such that $w_{s,1}\sigma(w_{s,1})=dw_{s,1}$, thus $w_{s,1}=\sigma(d)\in K(\xi)$ which is impossible since $[K(\xi)(w_{s,1})/K(\xi)]=2$ by assumption. Thus $w_{s,1}\sigma(w_{s,1})\in K$ and $[K(z\sigma(z))/K]=1$ and $z\sigma(z)\in K$. 2. (b)

If $f_{s}>1$,

[TABLE]

for $1\leq j\leq f_{s}-1$. Thus either $w_{s,j}\sigma(w_{s,j})$ is a Kummer generator of the degree $4$ subextension $K(w_{s,j}\sigma(w_{s,j}))/K(w_{s,j+2}\sigma(w_{s,j+2}))$ of $L/K$ (since any subextension of a cyclic extension is cyclic) or $w_{s,j}\sigma(w_{s,j})\in K(w_{s,j+1}\sigma(w_{s,j+1})$. But since $q\equiv-1\ mod\ 2^{e_{s}}$ thus $q\equiv-1\ mod\ 4$ and since $K$ does not contain a primitive $4^{th}$ root of unity $w_{s,j}\sigma(w_{s,j})$ cannot be a Kummer generator of degree $4$ and $w_{s,j}\sigma(w_{s,j})\in K(w_{s,j+1}\sigma(w_{s,j+1})$. So that, we can prove that

[TABLE]

with

[TABLE]

And using the same reasoning as in the case $(a)$ where $f_{s}=1$ we can deduce that $w_{s,e_{s}}\sigma(w_{s,e_{s}})\in K$ and finally, we have again $z\sigma(z)\in K$.

∎

Lemma 1.4**.**

*Let $L/K$ be a geometric cyclic extension of degree $\ell$ with $q\equiv-1\ mod\ \ell$. For any $z$ Kummer generator for $L(\xi)/K(\xi)$, there is $\eta\in\mathbb{F}_{q}(\xi)^{*}$ such that $\sigma(\eta z)+\eta z$ is a generator for $L/K$ and $\sigma(\eta)\eta=1$. *

Proof.

We write $z=z_{1}+\xi z_{2}$, we have thus that $z_{1},\ z_{2}\in L$ and $z_{2}\neq 0$ since $z\notin L$ and not both $z_{1}$ and $z_{2}$ are in proper the subextension of $L/K$ over $K$ since $z$ is a generator for $L(\xi)/K(\xi)$.

We have $\xi+\xi^{-1}\neq\pm 2$, since if this equality was satisfied, we would have $X^{2}-(\xi+\xi^{-1})X+1=X^{2}\mp 2X+1$ is not irreducible over $\mathbb{F}_{q}$ contradicting that $q\equiv-1\ mod\ \ell$ and thus $[\mathbb{F}_{q}(\xi):\mathbb{F}_{q}]=2$.

Taking $\zeta=\frac{\xi-\xi^{-1}}{(\xi+\xi^{-1})^{2}-4},$

[TABLE]

Taking $\zeta^{\prime}=\frac{\xi^{-2}-1}{(\xi+\xi^{-1})^{2}-4}$, we get

[TABLE]

For any $1\leq k\leq f_{i}$ and any $1\leq i\leq s$, we denote $L^{\ell_{i}^{k}}$ be the fixed field of $L$ by the unique subgroup of $Gal(L/K)$ isomorphic to $\mathbb{Z}/\ell_{i}^{k}\mathbb{Z}$. Since $z$ is a generator for $L(\xi)$ over $K(\xi)$, then $z$ is also a generator for $L(\xi)$ over $L^{\ell_{i}^{k}}(\xi)$, for any $1\leq k\leq f_{i}$ and any $1\leq i\leq s$. Given $1\leq i\leq s$, either $z_{1}$ or $z_{2}$ is a generator of $L/L^{\ell_{i}^{f_{i}}}$, indeed if that was not the case then both $z_{1}$ and $z_{2}$ would belong to a proper subextension of $L/L^{\ell_{i}^{f_{i}}}$ but since any such proper extension is contained in $L^{\ell_{i}^{f_{i}-1}}/K$. Then $z_{1}$ and $z_{2}$ would both belong to $L^{\ell_{i}^{f_{i}-1}}$ and $z=z_{1}+\xi z_{2}\in L^{\ell_{i}^{f_{i}-1}}(\xi)$ contradicting that $z$ is a generator for $L(\xi)/L^{\ell_{i}^{f_{i}}}(\xi)$.

That also implies in this case, that either $\mathfrak{z}_{1}=z+\sigma(z)$ or $\mathfrak{z}_{2}=\xi^{-1}z+\sigma(\xi^{-1}z)$ is a generator for $L/L^{\ell_{i}^{f_{i}}}$. Indeed, $z+\sigma(z)=2z_{1}+(\xi+\xi^{-1})z_{2}$ and $\xi^{-1}z+\sigma(\xi^{-1}z)=2z_{2}+(\xi+\xi^{-1})z_{1}$. As before, if none of them was a generator for $L/L^{\ell_{i}^{f_{i}}}$, we would have that they both belong to $L^{\ell_{i}^{f_{i}-1}}$ but then the same would be true for

[TABLE]

and for

[TABLE]

Contradicting that either $z_{1}$ or $z_{2}$ is a generator for $L/L^{\ell_{i}^{f_{i}}}$, since $\xi+\xi^{-1}\neq\pm 2$.

When $s=1$, the theorem is proven by the above.

Otherwise, when $s>1$, note that if we have $w\in L$ generator for $L$ over $L^{\ell_{i}^{f_{i}}}$ and $L^{\ell_{j}^{f_{j}}}$ for $i\neq j$. Then $[L:L^{\ell_{i}^{f_{i}}}]=[L^{\ell_{i}^{f_{i}}}(w):L^{\ell_{i}^{f_{i}}}]=\ell_{i}^{f_{i}}$ and $[L:L^{\ell_{j}^{f_{j}}}]=[L^{\ell_{j}^{f_{j}}}(w):L^{\ell_{j}^{f_{j}}}]=\ell_{j}^{f_{j}}$ and $\ell_{i}^{f_{i}}\ell_{j}^{f_{j}}|[L^{\ell_{i}^{f_{i}}\ell_{j}^{f_{j}}}(w):L^{\ell_{i}^{f_{i}}\ell_{j}^{f_{j}}}]$ where $L^{\ell_{i}^{f_{i}}\ell_{j}^{f_{j}}}$ is the fixed field of $L$ by the unique subgroup of $Gal(L/K)$ isomorphic to $\mathbb{Z}/(\ell_{i}^{f_{i}}\ell_{j}^{f_{j}})\mathbb{Z}$ and $L=L^{\ell_{i}^{f_{i}}\ell_{j}^{f_{j}}}(w)$. Thus, $\mathfrak{z}_{1}$ is a generator for $L/L^{I}$ and $\mathfrak{z}_{2}$ is a generator for $L/L^{J}$ where $I$ and $J$ are subsets of $\{1,\cdots,s\}$ such that $I\cup J=\{1,\cdots,s\}$ with $I$ and $J$ maximal subsets having this property and defining $L^{I}$ to be the fixed field of $L$ by the unique subgroup of $Gal(L/K)$ isomorphic to $\mathbb{Z}/(\prod_{i\in I}\ell_{i}^{f_{i}})\mathbb{Z}$ and $L^{J}$ to be the fixed field of $L$ by the unique subgroup of $Gal(L/K)$ isomorphic to $\mathbb{Z}/(\prod_{i\in J}\ell_{i}^{f_{i}})\mathbb{Z}$.

As a consequence, if $\mathfrak{z}_{1}=k\mathfrak{z}_{2}$ for some $k\in K^{*}$ then $\mathfrak{z}_{1}=z+\sigma(z)$ and $\mathfrak{z}_{2}=\xi^{-1}z+\sigma(\xi^{-1}z)$ are both generators for $L/K$ and the theorem is proven.

Also, if either $I$ or $J$ is the all $\{1,\cdots,s\}$ then either $\mathfrak{z}_{1}=z+\sigma(z)$ or $\mathfrak{z}_{2}=\xi^{-1}z+\sigma(\xi^{-1}z)$ is a generator for $L$ over $K$ and the theorem is again proven.

Otherwise, there is $i_{0}\in\{1,\cdots,s\}\backslash I$ and $i_{1}\in\{1,\cdots,s\}\backslash J$ and $i_{0}\neq i_{1}$. Note also that since by assumption $\ell=\prod_{i=1}^{s}\ell_{i}^{f_{i}}|q-1$ then $q-1\geq\ell>s$. Indeed, $\ell$ is bigger or equal to the product of the $s$ smallest prime numbers and the $s^{th}$ smallest prime number is already bigger than $s$. We denote $U_{q+1}$ the set of the $(q+1)^{th}$ roots of unity in $\mathbb{F}_{q}(\xi)^{*}$, we have $U_{q+1}=q+1>s+2$. Note that by assumption $\xi$ and $\xi^{-1}$ is a root of $R(X)=X^{2}-(\xi+\xi^{-1})X+1$ with $(\xi+\xi^{-1})\in\mathbb{F}_{q}$. Hence, we have also that $\xi^{q}$ is a root of $R(X)$ and thus either $\xi=\xi^{q}$ or $\xi^{-1}=\xi^{q}$ but since $\xi\notin\mathbb{F}_{q}$, we have $\xi^{-1}=\xi^{q}$. So that, for $\chi=c_{1}+\xi c_{2}\in U_{q+1}$, we have

[TABLE]

Moreover, $\mathfrak{z}_{1},\ \mathfrak{z}_{2}$ and $\mathfrak{z}_{1}+\chi\mathfrak{z}_{2}$ are distinct elements in $L$, for $\chi\in\mathbb{F}_{q}(\xi)^{*}$ and $\xi^{-1}\chi+1\in U_{q+1}$, since by assumption, $\mathfrak{z}_{1}\neq k\mathfrak{z}_{2}$, for any $k,l\in K$.

Let $\chi_{1}\in\mathbb{F}_{q}(\xi)^{*}$ such that $1+\chi_{1}\xi^{-1}\in U_{q+1}$ and $1\leq i\leq s$, then either $\mathfrak{z}_{1}$ or $\mathfrak{z}_{1}+\chi_{1}\mathfrak{z}_{2}$ is a generator for $L/L^{\ell_{i}^{f_{i}}}$, indeed, otherwise as before $\mathfrak{z}_{1}$ and $\mathfrak{z}_{1}+\chi_{1}\mathfrak{z}_{2}$ would both belong to $L^{\ell_{i}^{f_{i}-1}}$, and $\chi_{1}\mathfrak{z}_{2}=\mathfrak{z}_{1}+\chi_{1}\mathfrak{z}_{2}-\mathfrak{z}_{1}\in L^{\ell_{i}^{f_{i}-1}}$, then $\mathfrak{z}_{1}$ and $\mathfrak{z}_{2}$ would both belong to $L/L^{\ell_{i}^{f_{i}-1}}$ again contradicting that $z$ generates $L$ over $K$ as proven above. Thus, $\mathfrak{z}_{1}+\chi_{1}\mathfrak{z}_{2}$ is a generator of $L/L^{J_{1}}$ where $J_{1}$ is a subset of $\{1,\cdots,s\}$ such that $I\cup J_{1}=\{1,\cdots,s\}$ with $J_{1}$ maximal subset having this property and $L^{J_{1}}$ is the fixed field of $L$ by the unique subgroup of $Gal(L/K)$ isomorphic to $\mathbb{Z}/(\prod_{i\in J_{1}}\ell_{i}^{f_{i}})\mathbb{Z}$. Similarly, given $1\leq i\leq s$, we can prove that either $\mathfrak{z}_{2}$ or $\mathfrak{z}_{1}+\chi_{1}\mathfrak{z}_{2}$ is a generator for $L/L^{\ell_{i}^{f_{i}}}$ and thus $J\cup J_{1}=\{1,\cdots,s\}$.

If $J_{1}$ is all $\{1,\cdots,s\}$ then $\mathfrak{z}_{1}+\chi_{1}\mathfrak{z}_{2}=\eta z+\sigma(\eta z)$ is a generator $L/K$ where $\eta=1+\chi_{1}\xi^{-1}\in U_{q+1}$ thus $\eta\sigma(\eta)=1$, proving the theorem.

Otherwise, there is $i_{2}\in\{1,\cdots,s\}\backslash J_{1}$. But since $J\cup J_{1}=\{1,\cdots,s\}$ and $I\cup J_{1}=\{1,\cdots,s\}$ then $i_{0}$ and $i_{1}\in J_{1}$ thus $i_{0}$, $i_{1}$ and $i_{2}$ are distinct and $|J_{1}|\geq 2$.

We then let $\chi_{2}\in\mathbb{F}_{q}(\xi)^{*}$ such that $1+\chi_{2}\xi^{-1}\in U_{q+1}$ and $\chi_{1}\neq\chi_{2}$, as before we prove that $\mathfrak{z}_{1}+\chi_{2}\mathfrak{z}_{2}$ is also a generator of $L/L^{J_{2}}$ where $J_{2}$ is a subset of $\{1,\cdots,s\}$ such that $I\cup J_{2}=\{1,\cdots,s\}$, $L^{J_{2}}$ is the fixed field of $L$ by the unique subgroup of $Gal(L/K)$ isomorphic to $\mathbb{Z}/(\prod_{i\in J_{2}}\ell_{i}^{e_{i}})\mathbb{Z}$ with $I\cup J_{2}=\{1,\cdots,s\}$ and $J\cup J_{2}=\{1,\cdots,s\}$ with $J_{2}$ maximal subset having this property. Given $1\leq i\leq s$, we can also prove using the same argument as before that either $\mathfrak{z}_{1}+\chi_{1}\mathfrak{z}_{2}$ or $\mathfrak{z}_{1}+\chi_{2}\mathfrak{z}_{2}$ is a generator for $L/L^{\ell_{i}^{e_{i}}}$ since $\mathfrak{z}_{1}+\chi_{1}\mathfrak{z}_{2}-(\mathfrak{z}_{1}+\chi_{2}\mathfrak{z}_{2})=(\chi_{1}-\chi_{2})\mathfrak{z}_{2}$ and $\frac{1}{\chi_{1}}(\mathfrak{z}_{1}+\chi_{1}\mathfrak{z}_{2})-\frac{1}{\chi_{2}}(\mathfrak{z}_{1}+\chi_{2}\mathfrak{z}_{2})=\frac{\chi_{2}-\chi_{1}}{\chi_{1}\chi_{2}}\mathfrak{z}_{1}$. So that $J_{1}\cup J_{2}=\{1,\cdots,s\}$.

If $J_{2}$ is all $\{1,\cdots,s\}$ then $\mathfrak{z}_{1}+\chi_{2}\mathfrak{z}_{2}=\eta z+\sigma(\eta z)$ is a generator $L/K$ where $\eta=1+\chi_{2}\xi^{-1}$ proving the theorem.

Otherwise, there is $i_{3}\in\{1,\cdots,s\}\backslash J_{1}$, and since $I\cup J_{2}=J\cup J_{2}=J_{1}\cup J_{2}=\{1,\cdots,s\}$ we have $i_{0},i_{1},i_{2}\in J_{2}$ and thus $i_{0},i_{1},i_{2},i_{3}$ are all distinct and $|J_{2}|\geq 3$. Reproducing this process, with $\chi_{3},\cdots\chi_{s-1}\in\mathbb{F}_{q}(\xi)^{*}$ such that $1+\chi_{i}\xi^{-1}\in U_{q+1}$, for $1\leq i\leq s-1$, all distinct and distinct from $\chi_{1}$ and $\chi_{2}$ which exist since $s<|U_{q+1}|$, we find that either $\mathfrak{z}_{1}+\chi_{k}\mathfrak{z}_{2}$ is a generator for $L/K$ for some $k\in\{3,\cdots,s-2\}$ or $|J_{s-1}|\geq s$ and $\mathfrak{z}_{1}+\chi_{s-1}\mathfrak{z}_{2}$ is a generator for $L/K$. In any case, we find $\eta\in U_{q+1}$ such that $\eta z+\sigma(\eta z)$ is a generator for $L/K$

∎

Theorem 1.5**.**

Let $L/K$ be a geometric cyclic extension of degree $\ell$ with $q\equiv-1\ mod\ \ell$. There exists $w$ a Kummer generator for $L(\xi)/K(\xi)$ whose minimal polynomial is $X^{\ell}-a$ such that $y:=\sigma(w)+w$ is a generator for $L/K$ and $u:=\sigma(w)w\in K$ so that the minimal polynomial of $y$ over $K$ is $P^{\ell}_{u,\alpha}(X)$ as in lemma 1.1 where $\alpha=a+\sigma(a)$. Conversely, if $L/K$ is a geometric extension of degree $\ell$ with $q\equiv-1\ mod\ \ell$ and a generator $y$ whose minimal polymomial is of form $P^{\ell}_{u,\alpha}(X)$ where $u,\alpha\in K$ such that $u^{\ell}=a\sigma(a)$ and $\alpha=a+\sigma(a)$, for some $a\in K(\xi)\backslash K$ then $L/K$ is a cyclic extension.

Proof.

The first statement of the theorem is a direct consequence of Lemmas 1.1, 1.3 and 1.4, noting that if $z$ is a Kummer generator for $L(\xi)/K(\xi)$ then $\eta z$ is also a Kummer generator for $L(\xi)/K(\xi)$, by [1, Theorem 5.8.5].

Conversely, suppose that $L/K$ is an extension of degree $\ell$ with $q\equiv-1\ mod\ \ell$ and a generator $y$ whose minimal polymomial is of form $P^{\ell}_{u,\alpha}(X)$ where $u,\alpha\in K$ such that $u^{\ell}=a\sigma(a)$ and $\alpha=a+\sigma(a)$, for some $a\in K(\xi)$. Let $z_{1},z_{2}\in L^{alg}$ where $L^{alg}$ is the algebraic closure of $L$ the two roots of the polynomial $X^{2}-yX+u=0$. Then $y=z_{1}+z_{2}$ and $u=z_{1}z_{2}$, by Lemma 1.1, we have that

[TABLE]

And since $z_{2}=\frac{u}{z_{1}}$, we have

[TABLE]

and

[TABLE]

We have that

[TABLE]

then $a$ and $\sigma(a)$ are root of the polynomial

[TABLE]

Hence, up to reindexing, $z_{1}^{\ell}=a$ and $z_{2}^{\ell}=\sigma(a)$ and $\sigma(z_{1})\neq z_{1}$ since $a\in K(\xi)\backslash K$. Moreover, the coefficient of $T(X)=X^{2}-yX+u$ are in $L$, so that if $z_{1}$ is a root of $T(X)$ then $\sigma(z_{1})$ is a root of $T(X)$ and thus, $z_{2}=\sigma(z_{1})$. Since $\sigma(z_{1})\neq z_{1}$, we also obtain that $\xi\in L(z_{1})$ from $[L(z_{1}):L]\leq 2$ we can conclude that $L(z_{1})=L(\xi)$.

Since $z_{1}^{\ell}=a\in K(\xi)$, $[K(\xi)(z_{1}):K(\xi)]\leq\ell$. If $[K(\xi)(z_{1}):K(\xi)]<\ell$, since $K(\xi)(z_{1})=K(\xi)(\sigma(z_{1}))$, then $y:=z_{1}+\sigma(z_{1})$ would belong to a proper subextension of $L/K$, and $[K(y):K]<\ell$, which contradicts that $y$ is a generator for $L/K$. As a consequence, $z_{1}$ is a Kummer generator for $L(\xi)/K(\xi)$ and $L(\xi)/K(\xi)$ is cyclic. The set of all the $\zeta z_{1}$ where $\zeta$ is a $\ell^{th}$ root of unity is the set of distinct roots for the polynomial $X^{\ell}-a$ and then $\zeta z_{1}+\sigma(\zeta z_{1})$ are the distinct roots of $P^{\ell}_{u,\alpha}(X)$. Indeed, $\zeta=\xi^{i}$ for some $0\leq i\leq\ell-1$, thus $\sigma(\zeta)=\zeta^{-1}$, $\zeta z_{1}\sigma(\zeta z_{1})=u$ and $(\zeta z_{1})^{\ell}+\sigma(\zeta z_{1})^{\ell}=z_{1}^{\ell}+\sigma(z_{1})^{\ell}=\alpha$. Moreover, if $\zeta_{1}\neq\zeta_{2}$ $\ell^{th}$ root of unity, such that $\zeta_{1}z_{1}+\sigma(\zeta_{1}z_{1})=\zeta_{2}z_{1}+\sigma(\zeta_{2}z_{1})$ then $\frac{\zeta_{1}-\zeta_{2}}{-\sigma(\zeta_{1}-\zeta_{2})}=\frac{\sigma(z_{1})}{z_{1}}=\frac{1}{u}\sigma(z_{1})^{2}$, which contradicts that $L/K$ is a geometric extension. Proving that $\zeta z_{1}+\sigma(\zeta z_{1})$ are the distinct roots of $P^{\ell}_{u,\alpha}(X)$. This proves that $L/K$ is cyclic.

∎

Lemma 1.6**.**

Let $\sigma$ a generator of $Gal(K(\xi)/K)$ with $\xi$ a primitive $\ell^{th}$ root of unity. Let $d\in K(\xi)$, $\sigma(d)d=1$. There is $A,B\in\mathcal{O}_{K,x}$ such that

[TABLE]

where $C=\frac{A}{B}$.

When $K=\mathbb{F}_{q}(x)$, one can choose $(A,B)=1$.

Proof.

Let $\gamma\in K^{*}$. If we have $\gamma+d\sigma(\gamma)=0$, then $d=-\frac{\gamma}{\sigma(\gamma)}$. Noting that $\frac{\xi-\xi^{-1}}{\sigma(\xi-\xi^{-1})}=-1$. In this case, on can take $\theta=(\xi-\xi^{-1})\gamma$ and obtain

[TABLE]

Otherwise, let $\theta=\gamma+d\sigma(\gamma)\neq 0$, so that,

[TABLE]

We write $\theta=\theta_{1}+\xi\theta_{2}$, where $\theta_{1},\ \theta_{2}\in K$. One can write $\theta_{1}=\frac{A_{1}}{B_{1}}$ and $\theta_{2}=\frac{A_{2}}{B_{2}}$ where $A_{1},B_{1},A_{2},B_{2}\in\mathcal{O}_{K,x}$ so that $\theta=\frac{A_{1}}{B_{1}}+\xi\frac{A_{2}}{B_{2}}=\frac{A_{1}B_{2}+\xi A_{2}B_{1}}{B_{1}B_{2}}$ and $\sigma(\theta)=\frac{A_{1}B_{2}+\xi^{-1}A_{2}B_{1}}{B_{1}B_{2}}$, thus

[TABLE]

Taking $A=A_{1}B_{2}$ and $B=A_{2}B_{1}$, we obtain the theorem for general $K$.

When $K=\mathbb{F}_{q}(x)$, one can take $A=\frac{A_{1}B_{2}}{gcd(A_{1}B_{2},A_{2}B_{1})}$ and $B=\frac{A_{2}B_{1}}{gcd(A_{1}B_{2},A_{2}B_{1})}$ and prove the theorem, in this case. ∎

Corollary 1.7**.**

Let $L/K$ be a geometric cyclic extension of degree $\ell$ with $q\equiv-1\ mod\ \ell$. There exists $w$ a Kummer generator for $L(\xi)/K(\xi)$ whose minimal polynomial is $X^{\ell}-a$ such that $y:=\sigma(w)+w$ is a generator for $L/K$ and so that the minimal polynomial of $y$ over $K$ is $P^{\ell}_{u,\alpha}(X)$ as in lemma 1.1 where $\alpha=a+\sigma(a)$ and

when $\ell$ is an odd integer, $u=w\sigma(w)=a\sigma(a)=1$; 2. 2.

when $2|\ell$, \frac{a}{u^{\frac{\ell}{2}}}\sigma\big{(}\frac{a}{u^{\frac{\ell}{2}}}\big{)}=1 where $u=w\sigma(w)\in K$ and

[TABLE]

More precisely, $y$ generates the extension $L/L_{2}$ where $L_{2}$ is the fixed field of $L$ by the unique subgroup of $Gal(L/K)$ isomorphic to $\mathbb{Z}/2\mathbb{Z}$ with generating equation:

[TABLE]

where $\mathfrak{y}=\frac{\sigma(z)}{z}+\frac{z}{\sigma(z)}$ generates $L_{2}/K$ with minimal polynomial

[TABLE]

Proof.

Suppose $\ell$ is an odd integer.

Let $z$ be a Kummer generator for $L(\xi)/K(\xi)$ whose minimal polynomial is $X^{\ell}-c$, then $\frac{\sigma(z)}{z}$ is a Kummer generator $L(\xi)/K(\xi)$. Indeed, if it was not then $v=\frac{\sigma(z)}{z}$ would belong to a proper subextension $L^{\prime}$ of $L$ but since $z\sigma(z)\in K$ by Lemma 1.3, then $\frac{\sigma(z)}{z}z\sigma(z)=\sigma(z)^{2}\in L^{\prime}$ but since $z$ is a Kummer generator for $L(\xi)/K(\xi)$, so is $\sigma(z)$ and since $\ell$ is odd, $\sigma(z)^{2}$ is a Kummer generator of $L(\xi)/K(\xi)$, by [1, Proposition 5.8.7] which leads to a contradiction. By Lemma 1.4, there exist $\eta\in\mathbb{F}_{q}(\xi)^{*}$ such that $\eta\sigma(\eta)=1$, $\eta v+\sigma(\eta v)$ is a generator for $L/K$. Thus taking $w=\eta v$, we have $y:=w+\sigma(w)$ is a generator for $L/K$ and $w\sigma(w)=1$, $w^{\ell}=\eta^{\ell}\frac{\sigma(c)}{c}=:a$ and by Lemma 1.1, the minimal polynomial of $y$ is $P^{\ell}_{1,\alpha}$ where $\alpha=a+\sigma(a)$. 2. 2.

Suppose $\ell$ is an even integer. Thanks to Theorem 1.5, we can choose $w$ to be a Kummer generator for $L(\xi)/K(\xi)$ whose minimal polynomial is $X^{\ell}-a$ such that $y:=\sigma(w)+w$ is a generator for $L/K$ with minimal polynomial $P^{\ell}_{u,\alpha}(X)$ with $u=w\sigma(w)\in K$ and $\alpha=a+\sigma(a)$. Note that

[TABLE]

thus

[TABLE]

And

[TABLE]

[TABLE]

As a consequence,

[TABLE]

Note that since $P^{\ell}_{u,\alpha}(X)$ only involves even monomial we can write $P^{\ell}_{u,\alpha}(X)=Q(X^{2})$. where $Q(X)$ is a monic irreducible polynomial over $K$ and $deg(Q(X))=\frac{\ell}{2}$. Thus $Q(X)$ is the minimal polynomial of $y^{2}$, $[K(y^{2}):K]=\frac{\ell}{2}$ and $L_{2}=K(y^{2})$ and $\mathfrak{y}:=\frac{y^{2}-2u}{u}=\frac{\sigma(w)}{w}+\frac{w}{\sigma(w)}$ is also a generator for $L_{2}/K$ since $u\in K$. Moreover, $\frac{\sigma(w)}{w}$ Kummer generator for $L_{2}/K$ such that \big{(}\frac{\sigma(w)}{w}\big{)}^{\frac{\ell}{2}}=\frac{a}{u^{\frac{\ell}{2}}}, thus by Lemma 1.1, the minimal polynomial of $\mathfrak{y}$ is $P^{\frac{\ell}{2}}_{1,\frac{a}{u^{\frac{\ell}{2}}}}(X)$ and the the result.

∎

Combining together Theorem 1.5, Corollary 1.7 and Lemma 1.6, we obtain:

Corollary 1.8**.**

Let $L/K$ be a cyclic extension of degree $\ell$ with $q\equiv-1\ mod\ \ell$. There exists $w$ a Kummer generator for $L(\xi)/K(\xi)$ whose minimal polynomial is $X^{\ell}-a$ such that $y:=\sigma(w)+w$ is a generator for $L/K$ and $u:=\sigma(w)w\in K$ so that the minimal polynomial of $y$ over $K$ is

[TABLE]

for some $A,B\in\mathcal{O}_{K,x}$ such that $a=\frac{\sigma(A+\xi B)}{A+\xi B}$ and $u=1$, when $\ell$ is an odd integer; 2. 2.

[TABLE]

for some $A,B\in\mathcal{O}_{K,x}$ such that $a=u^{\frac{\ell}{2}}\frac{\sigma(A+\xi B)}{A+\xi B}$, when $\ell$ is an even integer;

where $c_{s,\ell}$ are defined in Lemma 1.1.

Remark 1.9**.**

Note that

[TABLE]

where $C=\frac{A}{B}$.

One can obtain the following result combining Theorem 1.5 and Proposition 5.8.7. VS

Lemma 1.10**.**

Let $L_{1}/K$ and $L_{2}/K$ be two geometric cyclic extensions of degree $\ell$ with $q\equiv-1\ mod\ \ell$. For $i=1,2$, there exists $w_{i}$ a Kummer generator for $L(\xi)/K(\xi)$ whose minimal polynomial is $X^{\ell}-a_{i}$ such that $y_{i}:=\sigma(w_{i})+w_{i}$ is a generator for $L/K$, so that the minimal polynomial of $y_{i}$ over $K$ is $P^{\ell}_{u_{i},\alpha_{i}}(X)$ as in lemma 1.1 where $\alpha_{i}=a_{i}+\sigma(a_{i})$ and $u_{i}:=\sigma(w_{i})w_{i}\in K$. Then, the following assertions are equivalent:

$L_{1}=L_{2}$** 2. 2.

$L_{1}(\xi)=L_{2}(\xi)$** 3. 3.

$w_{2}=cw_{1}^{j}$* so that $y_{2}=cw_{1}^{j}+\sigma(c)\sigma(w_{1})^{j}$ for all $1\leq j\leq n-1$ such that $(j,n)=1$ and $c\in K(\xi)$.* 4. 4.

$a_{2}=c^{\ell}a_{1}^{j}$* so that $\alpha_{2}=c^{\ell}a_{1}^{j}+\sigma(c^{\ell})\sigma(a_{1})^{j}$, for all $1\leq j\leq n-1$ such that $(j,n)=1$ and $c\in K(\xi)$. *

The following result is a direct corollary of the previous lemma.

Corollary 1.11**.**

Let $L/K$ be a geometric cyclic extension of degree $\ell$ with $q\equiv-1\ mod\ \ell$. There exists $w$ a Kummer generator for $L(\xi)/K(\xi)$ whose minimal polynomial is $X^{\ell}-a$ such that $y:=\sigma(w)+w$ is a generator for $L/K$, so that the minimal polynomial of $y$ over $K$ is $P_{\ell,u,\alpha}(X)$ as in lemma 1.1 where $\alpha=a+\sigma(a)$ and $u:=\sigma(w)w\in K$. Then the Galois group can be identified with the group of the $\ell^{th}$ root of unity, and the Galois action is given by $y\mapsto y_{\zeta}=\sigma(\zeta w)+\zeta w$

2 Ramification

Theorem 2.1**.**

*Let $L/K$ be a geometric cyclic extension of degree $\ell$ with $q\equiv-1\ mod\ \ell$. As in Theorem 1.5, we choose $w$ a Kummer generator for $L(\xi)/K(\xi)$ whose minimal polynomial is $X^{\ell}-a$ such that $y:=\sigma(w)+w$ is a generator for $L/K$, so that the minimal polynomial of $y$ over $K$ is $P_{\ell,u,\alpha}(X)$ as in lemma 1.1 where $\alpha=a+\sigma(a)$ and $u:=\sigma(w)w\in K$. Let $\mathfrak{p}$ be a place of $K$ and $\mathfrak{P}$ a place of $L$ above $\mathfrak{p}$. We denote $e(\mathfrak{P}|\mathfrak{p})$ the index of ramification of $\mathfrak{P}|\mathfrak{p}$, We have *

1. $\cdot$

when $v_{\mathfrak{p}}(\frac{u^{\ell}}{\alpha^{2}})\geq 0$ then

• $\cdot$

v_{\mathfrak{p}}\big{(}\alpha\big{)}\neq 0, $\mathfrak{p}$ is of even degree and $e(\mathfrak{P}|\mathfrak{p})=\frac{\ell}{(\ell,v_{\mathfrak{p}}(\alpha))}.$

• $\cdot$

v_{\mathfrak{p}}\big{(}\alpha\big{)}=0, $e(\mathfrak{P}|\mathfrak{p})=1$. 2. $\cdot$

when $v_{\mathfrak{p}}(\frac{u^{\ell}}{\alpha^{2}})<0$ then

• $\cdot$

$e(\mathfrak{P}|\mathfrak{p})=2$, if $(v_{\mathfrak{p}}(u),2)=1$.

• $\cdot$

$e(\mathfrak{P}|\mathfrak{p})=1$, otherwise.

When $\ell$ is odd, one can choose $y$ such that $u=1$, $\mathfrak{p}$ is unramified if and only if $v_{\mathfrak{p}}(\alpha)\geq 0$. Moreover, when $\mathfrak{p}$ is ramified ,$\mathfrak{p}$ is of even degree and

[TABLE]

Proof.

Let $\mathfrak{p}$ be a place of $K$, $\mathfrak{p}_{\xi}$ a place of $K(\xi)$ above $\mathfrak{p}$ and $\mathfrak{P}$ a place of $L$ above $\mathfrak{p}$ From [1, Theorem 5.8.12], since $K(\xi)/K$ is unramified and the index of ramification is multiplicative in tower, $\mathfrak{p}$ is unramified in $L$ if and only if $\mathfrak{p}_{\xi}$ is unramified in $L(\xi)$. That is, $\ell|v_{\mathfrak{p}_{\xi}}(a)$. Thus, in order to prove the theorem, we need to determine $v_{\mathfrak{p}_{\xi}}(a)$ in terms of $v_{\mathfrak{p}}(\alpha)$ and $v_{\mathfrak{p}}(u)$. For if, note that $a$ satisfies the equation

[TABLE]

since $\alpha=a+\sigma(a)$ and $u^{\ell}=a\sigma(a)$.

Moreover $K(a)=K(\xi)$ since $a\in K(\xi)\backslash K$, thus $K(a)/K$ is unramified.

We have

[TABLE]

When $v_{\mathfrak{p}}(\frac{u^{\ell}}{\alpha^{2}})<0$, we find by the triangular inequality for valuation that v_{\mathfrak{p}_{\xi}}\big{(}\frac{a}{\alpha}\big{)}<0 and 2v_{\mathfrak{p}_{\xi}}\big{(}\frac{a}{\alpha}\big{)}=v_{\mathfrak{p}}(\frac{u^{\ell}}{\alpha^{2}}) then

[TABLE]

When $v_{\mathfrak{p}}(\frac{u^{\ell}}{\alpha^{2}})=0$, by the triangular inequality, we obtain that v_{\mathfrak{p}_{\xi}}\big{(}\frac{a}{\alpha}\big{)}=0, thus

[TABLE]

When $v_{\mathfrak{p}}(\frac{u^{\ell}}{\alpha^{2}})>0$, then by Kummer theorem applied to the polynomial $X^{2}-X+\frac{u^{\ell}}{\alpha^{2}}$, we know that $\mathfrak{p}$ splits in $K(\xi)$, therefore, by [1, Theorem 6.2.1], we obtain that $\mathfrak{p}$ is of even degree. Moreover, by the triangular inequality v_{\mathfrak{p}_{\xi}}\big{(}\frac{a}{\alpha}\big{)}\geq 0, for any place $\mathfrak{p}_{\xi}$ of $K(\xi)$ over $\mathfrak{p}$. More precisely, either v_{\mathfrak{p}_{\xi}}\big{(}\frac{a}{\alpha}\big{)}=0 or v_{\mathfrak{p}_{\xi}}\big{(}\frac{a}{\alpha}\big{)}=v_{\mathfrak{p}}\big{(}\frac{u^{\ell}}{\alpha^{2}}\big{)}. Moreover,

[TABLE]

Thus,

[TABLE]

From this we deduce that either

[TABLE]

that is,

[TABLE]

or

[TABLE]

that is,

[TABLE]

This permitting to obtain the desire result. ∎

Remark 2.2**.**

When $\ell$ is odd, one can choose a single place $\mathfrak{P}_{\infty}$ at infinity in $K$ such that $v_{\mathfrak{P}_{\infty}}(a)\geq 0$ and thus $\mathfrak{P}_{\infty}$ is unramified. (see [4, Corollary 3.12])

Lemma 2.3**.**

Let $\ell$ be an odd integer. Let $L/K$ be a cyclic extension of degree $\ell$ with $q\equiv-1\ mod\ \ell$. Given $\mathfrak{p}$ a place of $K$. There is $w$ a Kummer generator for $L(\xi)/K(\xi)$ whose minimal polynomial is $X^{\ell}-a$ such that $y:=\sigma(w)+w$ is a generator for $L/K$, so that the minimal polynomial of $y$ over $K$ is $P^{\ell}_{1,\alpha}(X)$ as in lemma 1.1 where $\alpha=a+\sigma(a)$, $\sigma(w)w=1$ and either $v_{\mathfrak{p}}(\alpha)=-m$ where $0\leq m\leq\ell-1$ or $v_{\mathfrak{p}}(\alpha)\geq 0$. Moreover, when $v_{\mathfrak{p}}(\alpha)=-m$ where $0\leq m\leq\ell-1$ then $\mathfrak{p}$ is ramified and $v_{\mathfrak{p}}(\alpha)\geq 0$ then $\mathfrak{p}$ is unramified.

Proof.

Let $\mathfrak{p}$ a place of $K$. By Corollary 1.8, there is $v$ a Kummer generator for $L(\xi)/K(\xi)$ whose minimal polynomial is $X^{\ell}-c$ such that $z:=\sigma(v)+v$ is a generator for $L/K$ and $\sigma(v)v=1$ so that the minimal polynomial of $z$ over $K$ is $P^{\ell}_{1,\beta}(X)$ as in lemma 1.1 where $\beta=c+\sigma(c)$. From the proof of the previous lemma 2.1, we have that either $v_{\mathfrak{p}}(\beta)\geq 0$ when $v_{\mathfrak{p}_{\xi}}(c)=0$ or $\mathfrak{p}$ split $K(\xi)$ and for one of the place $\mathfrak{p}_{\xi}$ above $\mathfrak{p}$, we have

[TABLE]

By Lemma 1.6, there is $\gamma\in K(\xi)^{*}$ such that

[TABLE]

We write $v_{\mathfrak{p}_{\xi}}(\gamma)=\ell s+m$ where $0\leq m\leq\ell-1$, using weak approximation theorem we choose $\lambda\in K(\xi)$ such that $v_{\mathfrak{p}_{\xi}}(\lambda)=-s$. So that $\frac{\sigma(\lambda)}{\lambda}v$ is a Kummer generator for $L(\xi)/K(\xi)$ whose minimal polynomial is $X^{\ell}-\lambda^{\ell}c$ and

[TABLE]

By Lemma 1.4, there is $\eta\in\mathbb{F}_{q}(\xi)^{*}$ such that $\sigma(\eta)\eta=1$, $w=\eta\frac{\sigma(\lambda)}{\lambda}v$ is a Kummer generator $L(\xi)/K(\xi)$ with minimal polynomial $X^{\ell}-a$ where a=\eta^{\ell}\big{(}\frac{\sigma(\lambda)}{\lambda}\big{)}^{\ell}c and $y:=w+\sigma(w)$ is a generator for $L/K$ and its minimal polynomial is $P^{\ell}_{1,\alpha}(X)$ with $\alpha=a+\sigma(a)$ and $v_{\mathfrak{p}}(\alpha)=-m$.

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Lemma 2.4**.**

Let $\ell$ be an odd integer. Suppose $K=\mathbb{F}_{q}(x)$. Let $L/K$ be a cyclic extension of degree $\ell$ with $q\equiv-1\ mod\ \ell$. There is $w$ a Kummer generator for $L(\xi)/K(\xi)$ whose minimal polynomial is $X^{\ell}-a$ such that $y:=\sigma(w)+w$ is a generator for $L/K$ so that the minimal polynomial of $y$ over $K$ is $P^{\ell}_{1,\alpha}(X)$ as in lemma 1.1 where $\alpha=a+\sigma(a)$ and $\sigma(w)w=1$ such that at any $\mathfrak{p}$ place of $K$ either $v_{\mathfrak{p}}(\alpha)=-m$ where $0\leq m\leq\ell-1$ or $v_{\mathfrak{p}}(\alpha)\geq 0$. Moreover, when $v_{\mathfrak{p}}(\alpha)=-m$ where $0\leq m\leq\ell-1$ then $\mathfrak{p}$ is ramified, $v_{\mathfrak{p}}(\alpha)\geq 0$ then $\mathfrak{p}$ is unramified and $v_{\mathfrak{p}_{\infty}}(\beta)\geq 0$ where $\mathfrak{p}_{\infty}$ is the pole divisor of $x$ so that $\mathfrak{p}_{\infty}$ is unramified.

Proof.

By Corollary 1.8, there is $w$ a Kummer generator for $L(\xi)/K(\xi)$ whose minimal polynomial is $X^{\ell}-c$ such that $z:=\sigma(v)+v$ is a generator for $L/K$ so that the minimal polynomial of $z$ over $K$ is $P^{\ell}_{1,\beta}(X)$ as in lemma 1.1 where $\beta=c+\sigma(c)$ and $\sigma(v)v=1$. By Lemma 1.6, there is $\gamma\in F_{q}(\xi)[x]^{*}$ such that $(\sigma(\gamma),\gamma)=1$ and

[TABLE]

Since $F_{q}(\xi)[x]$ is a unique factorization domain and we write

[TABLE]

where $\gamma_{i}$ are distincts irreducible polynomials in $\mathbb{F}_{q}[x]$ and $e_{i}=\ell s_{i}+m_{i}$ where $0\leq m_{i}\leq\ell-1$, for $0\leq i\leq t$. Since $(\sigma(\gamma),\gamma)=1$ we have $\sigma(\gamma_{i})\neq\gamma_{i}$.

Let $\lambda=\prod_{i=1}^{t}\gamma_{i}^{-s_{i}}$ so that $\theta=\lambda^{\ell}\gamma=\prod_{i=1}^{t}\gamma_{i}^{m_{i}}$ and

[TABLE]

Moreover, $\frac{\sigma(\lambda)}{\lambda}v$ is a Kummer generator for $L(\xi)\ K(\xi)$. By Lemma 1.4, there is $\eta\in\mathbb{F}_{q}(\xi)^{*}$ such that $\sigma(\eta)\eta=1$, $w=\eta\frac{\sigma(\lambda)}{\lambda}v$ is a Kummer generator $L(\xi)/K(\xi)$ with minimal polynomial $X^{\ell}-a$ where a=\eta^{\ell}\big{(}\frac{\sigma(\lambda)}{\lambda}\big{)}^{\ell}c and $y:=w+\sigma(w)$ is a generator for $L/K$ and its minimal polynomial is $P^{\ell}_{1,\alpha}(X)$ with

[TABLE]

$\gamma_{i}\sigma(\gamma_{i})$ is then an irreducible polynomial in $\mathbb{F}_{q}[x]$ and $(\theta\sigma(\theta),\theta^{2}+\sigma(\theta)^{2})=1$ so that if $\mathfrak{p}_{i}$ is the finite place of $K$ corresponding to $\gamma_{i}\sigma(\gamma_{i})$, $v_{\mathfrak{p}_{i}}(a+\sigma(a))=-m_{i}$. Clearly, at any other finite place, $v_{\mathfrak{p}_{i}}(a+\sigma(a))\geq 0$.

Finally, since $deg(\theta)=deg(\sigma(\theta))$, we have $v_{\mathfrak{p}_{\infty}}(a)=0$, so that $v_{\mathfrak{p}_{\infty}}(a+\sigma(a))\geq 0$, concluding the proof of the lemma.

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