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I'm reading a brief introduction on Galois theory, it talks about splitting field. The idea is genius, however I have some doubts on whether in general a splitting field always exists?

By definition, a splitting field of a polynomial $p(X)$ over a field $K$ is a field extension $L$ of $K$ over which $p$ factors into linear factors $p(X) = \prod_{i=1}^{deg(p)} (X-a_i)$ where for each $i$ we have $(X - a_i) \in L[X]$ and such that the roots $a_i$ generate $L$ over $K$.

If $K$ is $\mathbb R$, the result is straightforward due to http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra as it states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

But if $K$ is a general field, does such an extension $L$ always exist? How to prove that?

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  • Start with Kronecker's theorem:

    For every non-constant polynomial $p$ over $K$, there is an extension $E$ of $K$ such that $p$ has a root in $E$.

$\qquad$ One can simply take $E=K[X]/(q)$, where $q$ is an irreducible factor of $p$.

  • Remove the roots of $p$ in $E$ from $p$ and repeat.

  • Since the degrees of the polynomial decrease steadily, the process stops after at most $n$ steps, where $n$ is the degree of $p$.

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