Let $F$ be a field and $K$ a splitting field for some nonconstant polynomial over $F$. Show that $K$ is a finite extension of $F$.

Question

help me with this excercice..

Let $F$ be a field and $K$ a splitting field for some nonconstant polynomial over $F$. Show that $K$ is a finite extension of $F$.

I try

$K$ is a splitting field.

$K=F(a_1,a_2,...,a_n)$ for $a_i$ roots for polynomial over $F$, i guess that have proof $[K:F]=n$ help

Answer 1

Hint:

First prove it for one root and use induction for $n$ roots. Suppose $K=F(a)$ and $|K:F|=m$. Then for any $x\in K$, $1, x, \cdots, x^m$ must be linear dependent because $K$ is vector space with dimension of $m$.