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
Suppose first that
is simple, that is to say
( α )
α ∈
be any intermediate field between
be the minimal polynomial of
be the field extension of
generated by all the coefficients of
by definition of the minimal polynomial, but the degree of
) simply the degree of
Therefore, by multiplicativity of degree,
is the minimal polynomial of
, and since there are only finitely many divisors of
, the first direction follows.
Conversely, if the number of intermediate fields between
is finite, we distinguish two cases: This theorem was found and proven in 1910 by Ernst Steinitz.