Steinitz's theorem (field theory)

In field theory, Steinitz's theorem states that a finite extension of fields $$L/K$$ is simple if and only if there are only finitely many intermediate fields between $$K$$ and $$L$$.

Proof
Suppose first that $$L/K$$ is simple, that is to say $$L = K(\alpha)$$ for some $$\alpha \in L$$. Let $$M$$ be any intermediate field between $$L$$ and $$K$$, and let $$g$$ be the minimal polynomial of $$\alpha$$ over $$M$$. Let $$M'$$ be the field extension of $$K$$ generated by all the coefficients of $$g$$. Then $$M' \subseteq M$$ by definition of the minimal polynomial, but the degree of $$L$$ over $$M'$$ is (like that of $$L$$ over $$M$$) simply the degree of $$g$$. Therefore, by multiplicativity of degree, $$[M:M'] = 1$$ and hence $$M = M'$$.

But if $$f$$ is the minimal polynomial of $$\alpha$$ over $$K$$, then $$g | f$$, and since there are only finitely many divisors of $$f$$, the first direction follows.

Conversely, if the number of intermediate fields between $$L$$ and $$K$$ is finite, we distinguish two cases:


 * 1) If $$K$$ is finite, then so is $$L$$, and any primitive root of $$L$$ will generate the field extension.
 * 2) If $$K$$ is infinite, then each intermediate field between $$K$$ and $$L$$ is a proper $$K$$-subspace of $$L$$, and their union can't be all of $$L$$. Thus any element outside this union will generate $$L$$.

History
This theorem was found and proven in 1910 by Ernst Steinitz.