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