有限域：一个定理的证明

rickyin

给出费马小定理：
Lemma 1. In number theory, Fermat's little theorem states that if p is a prime number, then for any integer \(a\), the number \(a^p-a\) is an integer multiple of \(p)\) In the notation of modular arithmetic, this is expressed  \(a^p \equiv a \mod p\).

于是可得以下Theorem 2.11.
Theorem 2.11. Let  \(f(X)\) be a polynomial over  \(GF(q)\). Let \(\beta\) be an element of the extension field  \(GF(q^m)\) of \(GF(q)\). If  \(\beta\) is a root of  \(f(X)\), then, for any non-negative integer  \(t\),  \(\beta^{q^t}\) is also a root of  \(f(X)\).

Proof. Based on Lemma 1 we have that  \([f(X)]^{q^t} = f(X^{q^t}) = f(X)\). Then, it is obvious.

其中 \([f(X)]^{q^t} = f(X^{q^t}) = f(X)\) 是由费马小定理得出，但我不确定我的判断是否正确，希望大家能够批评指正。

rickyin

无法由费马小定理得出，因为\(q^t \)不是素数，因此并不能用费马小定理进行证明，而是使用环的性质进行证明。具体可见《Finite Fields》1.46. Theorem