1. ## extension

Let $E$ be an algebraic extension of a field $F$, and let $\sigma$ be an automorphism of $E$ leaving $F$ fixed. Let $\alpha \in E$. Show that $\sigma$ induces a permutation of the set of zeros of $\text{irr}(\alpha, F)$ that are in $E$.

Let $\text{irr}(\alpha, F) = a_0+a_{1}x+ \cdots + a_{n}x^n$. Then $a_0+ a_{1}\alpha + \cdots + a_{n} \alpha^{n} = 0$. So $a_0+ a_1 \sigma(\alpha) + \cdots + a_{n} \sigma(\alpha)^{n} = \sigma(0) = 0$. So $\sigma(\alpha)$ is a zero of $\text{irr}(\alpha, F)$.

2. Originally Posted by Sampras
Let $E$ be an algebraic extension of a field $F$, and let $\sigma$ be an automorphism of $E$ leaving $F$ fixed. Let $\alpha \in E$. Show that $\sigma$ induces a permutation of the set of zeros of $\text{irr}(\alpha, F)$ that are in $E$.

Let $\text{irr}(\alpha, F) = a_0+a_{1}x+ \cdots + a_{n}x^n$. Then $a_0+ a_{1}\alpha + \cdots + a_{n} \alpha^{n} = 0$. So $a_0+ a_1 \sigma(\alpha) + \cdots + a_{n} \sigma(\alpha)^{n} = \sigma(0) = 0$. So $\sigma(\alpha)$ is a zero of $\text{irr}(\alpha, F)$.
correct. you should probably explain it a little bit more if it's a part of your homework.