Steinitz's theorem (field theory) explained

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.

1. 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 .^[1]

History

This theorem was found and proven in 1910 by Ernst Steinitz.^[2]

References

1. https://stacks.math.columbia.edu/tag/030N Lemma 9.19.1 (Primitive element)
2. Steinitz. Ernst. 1910. Algebraische Theorie der Körper.. Journal für die reine und angewandte Mathematik. de. 1910. 137 . 167–309. 10.1515/crll.1910.137.167. 120807300 . 1435-5345.