Artin-Schreier Extensions

Subjects: Field Theory
Links: Finite Fields, Splitting Fields and Normal Field Extensions, Characteristic of a Ring, Galois Field Extensions, Solvable Polynomials and Radical Extensions

Let $K$ be a field of characteristic $p>0$, and $E/K$ a cyclic extension of order $p^{m-1}$ with $m>1$.

Def: An Artin-Schreier polynomial $A_{\alpha }(x)\in K[x]$ is of the form $$x^p-x-\alpha,$$with $\alpha \in K$.

Obs: An almost immediate property of Artin-Schreier polynomials that we can notice is the equation $$A_\alpha(x+y) = A_\alpha(x)+ A_\alpha(y)- A_\alpha(0) $$

Prop: If $A_{\alpha }(x)$ has a root in $K$, then all roots of $A_{\alpha }(x)$ is in $K$. Otherwise, $A_{\alpha }(x)$ is irreducible over $K$. In this case, let $\theta$ be a root of $A_{\alpha }(x)$, then $K(\theta )/K$ is a cyclic extension of degree $p$. We see that $K(\theta )$ is the splitting field of $A_{\alpha }(x)$, the generator of $\text{Gal}(K(\theta )/K)$ is $\sigma (x)=x+1$.

Def: The field extension $E/K$ is called an Artin-Schreier extension if $E=K(\theta )$ for some $\alpha \in L$.

Cor: If $A_{\alpha }(x)$ is an Artin-Schreier polynomial in ${\mathbb{F}}_p$, and $\alpha \in {\mathbb{F}}_p^{\times }$, then ${\mathbb{F}}_p(\theta )\cong {\mathbb{F}}_{p^p}$.

Let's consider the case where $L/K$ is a cyclic extension of degree $n$ such that $\text{char }K\mid n$.

Prop: Let $L/K$ be a cyclic extension of degree $n$, where $n=p^{k}m$, $\text{char K}=p>0$, and $p\nmid m$, then there exists a sequence of extensions $$K \subseteq M_k \subseteq\cdots\subseteq M_0 \subseteq L $$such that $L/M_0$ is a cyclic extension of degree $m$, and foro $i\in \{1,\dots ,k\}$ $M_i/M_{i+1}$ is a cyclic extension of degree $p$.

With this proposition we only need to consider the case where $n=p=\text{char K}=p$.

Artin-Schreier Theorem: Let $K$ be a field of characteristic $p\ne 0$.

• If $L/K$ is cyclic extension of degree $p$, then $L$ is the splitting field of a an irreducible polynomial $x^p-x-a\in K[x]$. In fact, $L=K(\alpha )$ where $\alpha$ is any root of $x^p-x-a$.
• Reciprocally, if $L$ is the splitting field of the polynomial $x^p-x-a\in K[x]$, then $L/K$ is cycic. Furthermore,
  • The polynomial $x^p-x-a$ has all of its roots in $K$, so $L=K$, or
  • The polynomial $x^p-x-a$ is irreducible over $K$, and thus $L/K$ is cyclic of degree $p$, and $L=K(\alpha )$, with $\alpha$ is any root of $x^p-x-a$.