Let K denote the splitting field of a polynomial f(x) over ℚ and let G denote the Galois group of K: ℚ. Then the degree of K: ℚ equals the order of G. Further for every subgroup H of G we can define its fixed field

which is a subfield of K. And for every subfield F of K,

is a subgroup of G. Then ^* and ^† are inverses of one another. Further N is a normal subgroup of G if and only if N^† is the splitting field of some polynomial. (With further technical additions the correspondence can be extended to base fields other than ℚ.)