Kronecker’s Theorem

Theorem
Let $K$ be a field and $f$ a polynomial over $K$ of strictly positive degree.

Then there exists a finite extension $F/K$ in which $f$ has at least one root.

Proof
We have that the ring of polynomial forms over a field is a principal ideal domain.

Therefore it is also a unique factorisation domain.

That is, we can write $f = u g_1\cdots g_r$ with $u$ a unit and $g_i$ irreducible for $i=1,\ldots,r$.

Clearly it is sufficient to find an extension of $K$ in which one of the irreducible factors of $f$ has a zero.

Let $K[X]$ be the ring of polynomial functions in the variable $X$ over $K$.

Let $F = K[X] / \langle g_1 \rangle$, where $\langle g_1 \rangle$ is the ideal generated by $g_1$.

By principal ideal of irreducible element $\langle g_1 \rangle$ is maximal.

Therefore by maximal ideal iff quotient ring is a field, $F$ is a field.

Moreover:

So $X + \langle g_1 \rangle$ is a zero of $g_1$ in $F$.

We are done.