matematicaeducativa.com The Galois Correspondence 115 Lemma 8.5. If H is a subgroup of Î(L: K), then Ht is a subfield of L containing K Prof. Let x,YEHt , and α E H. Then α (x + y) = α(x) + α(y) = x + y so x +yEH. Similarly H is closed under subtraction, multiplication, and division (by nonzero elements), so H is a subfield of L. Since a ET(L: K) we have (k)-k for all ke K, so K CH Definition 8.6. With the above notation, H is the fixed field of H It is easy to see that like *, the map t reverses inclusions: if H C Gthen 2G It is also easy to verify that if M is an intermediate field and H is a subgroup of the Galois group, then MCM (8.4) Indeed, every element of M is fixed by every automorphism that fixes all of M, and every element of H fixes those elements that are fixed by all of H. Example 8.4(2) shows that these inclusions are not always equalities, for there If we letå¤denote the set of intermediate fields, and y the set of subgroups of the Galois group, then we have defined two maps which reverse inclusions and satisfy equation (8.4). These two maps constitute the Galois correspondence between and . Galois's results can be interpreted as giv ing conditions under which and are mutual inverses, setting up a bijection between and . The extra conditions needed are called separability (which is automatic over C) and normality. We discuss them in Chapter 9
