# extension

• Jan 9th 2010, 06:40 AM
Sampras
extension
Let $\displaystyle E$ be an algebraic extension of a field $\displaystyle F$, and let $\displaystyle \sigma$ be an automorphism of $\displaystyle E$ leaving $\displaystyle F$ fixed. Let $\displaystyle \alpha \in E$. Show that $\displaystyle \sigma$ induces a permutation of the set of zeros of $\displaystyle \text{irr}(\alpha, F)$ that are in $\displaystyle E$.

Let $\displaystyle \text{irr}(\alpha, F) = a_0+a_{1}x+ \cdots + a_{n}x^n$. Then $\displaystyle a_0+ a_{1}\alpha + \cdots + a_{n} \alpha^{n} = 0$. So $\displaystyle a_0+ a_1 \sigma(\alpha) + \cdots + a_{n} \sigma(\alpha)^{n} = \sigma(0) = 0$. So $\displaystyle \sigma(\alpha)$ is a zero of $\displaystyle \text{irr}(\alpha, F)$.
• Jan 9th 2010, 07:26 PM
NonCommAlg
Quote:

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

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

correct. you should probably explain it a little bit more if it's a part of your homework.