I have been reading about fields and I would like assistance with the following:
Let $k$ be an arbitrary field and $f(x)$ be an irreducible polynomial in $k[x]$. Then:
Prop: There exists a field $K$ containing $k$ and an element $\alpha\in K$ such that $f(\alpha)=0$.
I want to show that given a field $k$ and a polynomial $f(x)\in k[x]$ there is a field $K \supset k$ such that $[K:k]$ is finite and $f(x)=(x-\alpha_{1})(x-\alpha_{2})\cdots (x-\alpha_{n})$ in $K[x]$. Thanks.