Let E be algebraic extension of F, an embedding maps a root of minimal polynomial to a root of minimal polynomial

Question

Let E be algebraic extension of F, and let σ:E⟶E be an embedding of E into itself over F.

Let a ∈ E and let p(x) be its minimal polynomial over F.

My book says, without any comments, that σ must map a root of p(x) onto a root of p(x), but I don't understand why this should be the case.

Please help! Thanks in advance :)

Answer 1

Recall that $\sigma$ is a ring homomorphism (that means that $\sigma(1)=1$ and $\sigma(x+y)=\sigma(x)+\sigma(y)$ and $\sigma(xy) = \sigma(x)\sigma(y)$ for any $x$ and $y$ in $E$) such that $\sigma(f)=f$ if $f \in F$.

Now, if you write $p(x) \in F[x]$ as $\sum_{k\geq0} f_kx^k$, apply $\sigma$ on both sides to the equation $\sum_{k\geq0} f_ka^k=0$ and see what happens.