Galois correspondence
The Galois correspondence is a dictionary translating questions about fields into questions about groups. The fields sandwiched between K and a Galois extension L form a lattice; the subgroups of the Galois group form another lattice; and the correspondence matches them up perfectly, but upside-down — the biggest field talks to the smallest group and vice versa.
Concretely, for a finite Galois extension L/K with G = Gal(L/K), there is an inclusion-reversing bijection between intermediate fields F (with K subset of F subset of L) and subgroups H of G. It sends F to the subgroup Gal(L/F) of automorphisms fixing F, and sends H to its fixed field L^H. These two maps are mutually inverse: Gal(L/L^H) = H and L^{Gal(L/F)} = F. Moreover degrees and indices match, [L : F] = |Gal(L/F)| and [F : K] = [G : Gal(L/F)].
The correspondence also tracks normality: an intermediate field F is itself a Galois (equivalently normal) extension of K exactly when Gal(L/F) is a normal subgroup of G, and in that case Gal(F/K) is isomorphic to the quotient G / Gal(L/F). This is the precise sense in which solving towers of fields becomes a problem about subnormal series of groups — the bridge that connects radical solvability of polynomials to solvability of groups.
For L = Q(sqrt(2), sqrt(3)) over Q with G = Z/2Z x Z/2Z, the three order-2 subgroups correspond to the three quadratic subfields Q(sqrt(2)), Q(sqrt(3)), Q(sqrt(6)).
Three subgroups of the Klein four-group match three intermediate fields.
For infinite Galois extensions the bijection is restored only after restricting to closed subgroups in the Krull (profinite) topology; arbitrary subgroups need not be of the form Gal(L/F).