Steinitz's theorem (field theory)

In field theory, Steinitz's theorem states that a finite extension of fields [math]\displaystyle{ L/K }[/math] is simple if and only if there are only finitely many intermediate fields between [math]\displaystyle{ K }[/math] and [math]\displaystyle{ L }[/math].

Proof

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

But if [math]\displaystyle{ f }[/math] is the minimal polynomial of [math]\displaystyle{ \alpha }[/math] over [math]\displaystyle{ K }[/math], then [math]\displaystyle{ g | f }[/math], and since there are only finitely many divisors of [math]\displaystyle{ f }[/math], the first direction follows.

Conversely, if the number of intermediate fields between [math]\displaystyle{ L }[/math] and [math]\displaystyle{ K }[/math] is finite, we distinguish two cases:

1. If [math]\displaystyle{ K }[/math] is finite, then so is [math]\displaystyle{ L }[/math], and any primitive root of [math]\displaystyle{ L }[/math] will generate the field extension.
2. If [math]\displaystyle{ K }[/math] is infinite, then each intermediate field between [math]\displaystyle{ K }[/math] and [math]\displaystyle{ L }[/math] is a proper [math]\displaystyle{ K }[/math]-subspace of [math]\displaystyle{ L }[/math], and their union can't be all of [math]\displaystyle{ L }[/math]. Thus any element outside this union will generate [math]\displaystyle{ L }[/math].

History

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