域论(有限域)

本节默认$p$是素数

前置

• 形式导数

定义

• An element $\alpha$ of a field is an $n$th root of unity if ${\alpha }^n=1$. It is a primitive $n$th root of unity if ${\alpha }^n=1$ and ${\alpha }^m\ne 1$ for $0<m<n$.

性质

1. Let $E$ be a finite extension of degree $n$ over a finite field $F$. If $F$ has $q$ elements, then $E$ has $q^n$ elements.
2. If $E$ is a finite field of characteristic $p$, then $E$ contains exactly $p^n$ elements for some positive integer $n$.
3. Let $E$ be a field of $p^n$ elements contained in an algebraic closure $\overline{{\mathbb{Z}}_p}$ of ${\mathbb{Z}}_p$. The elements of $E$ are precisely the zeros in $\overline{{\mathbb{Z}}_p}$ of the polynomial $x^{p^n}-x$ in ${\mathbb{Z}}_p[x]$.
4. A finite extension $E$ of a finite field $F$ is a simple extension of $F$
5. If $F$ is a field of prime characteristic $p$ with algebraic closure $F$, then $x^{p^n}-x$ has $p^n$ distinct zeros in $F$. (使用形式导数证明)