Steinitz's theorem (field theory)

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

Proof

Suppose first that is simple, that is to say for some . Let be any intermediate field between and , and let be the minimal polynomial of over . Let be the field extension of generated by all the coefficients of . Then by definition of the minimal polynomial, but the degree of over is (like that of over ) simply the degree of . Therefore, by multiplicativity of degree, and hence .

But if is the minimal polynomial of over , then , and since there are only finitely many divisors of , the first direction follows.

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

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

History

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

References