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:
If is finite, then so is , and any primitive root of will generate the field extension. 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 .Lemma 9.19.1 (Primitive element), The Stacks project. Accessed on line July 19, 2023.
History This theorem was found and proven in 1910 by Ernst Steinitz.
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