Galois Closures for Rings by Owen Biesel

By Owen Biesel

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Bij(S, S) is the set of automorphisms of K 45 to the identity on N , namely GN , so we have a group homomorphism GK /GN ! G0 ; the fundamental theorem of Galois theory for infinite extensions (see, for example, [7, Thm. 3]) tells us that GN is an open finite-index subgroup of GK and that GK /GN ! G0 is an isomorphism, giving G0 a continuous GK -action. ¯ is a bijection 3. Note that N is its own normal closure; composition with N ,! K ⇠ ¯ The action of GK by post-composition is the G0 = AutK (N ) !

2 G-closures of ´ etale extensions are ´ etale In this section, we prove that if A is a degree-n extension of R which is ´etale as an R-algebra, then each G-closure of A over R is an ´etale degree-|G| extension of R. We 40 use the following lemma, whose present formulation is that of [2, Lem. 1. Let R be a ring, and let B be an R-algebra that is finitely generated as an R-module. Then B is an ´etale degree-n extension of R if and only if there exists an ´etale R-algebra C such that the morphism of schemes Spec(C) !

Sk and R ! R[x1 , . . ,xn ]G is injective. 3, since |G| is a non-zerodivisor the latter condition is guaranteed to hold. 4 An-closures In this section we give generators and relations for R[x1 , . . , xn ]An as an R[x1 , . . , xn ]Sn algebra, and combine this presentation with our earlier results to parametrize An closures of monogenic extensions. 1. Let n be a natural number. Then Z[x1 , . . , xn ]An is a Z[x1 , . . , xn ]Sn Q module with free basis {1, }, where is the An -orbit sum of ni=1 xii 1 .

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