By Owen Biesel
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Additional resources for Galois Closures for Rings
Sample text
1 G-closures of extensions with degree 2 In this section, we analyze extensions of degree at most two and their G-closures. First, note that the only degree-0 extension is the zero algebra R ! 0, because the R-module R0 is the zero module. Its S0 -closure is R, however, because 0⌦0 ⇠ = R. In general, letting 1 be the trivial subgroup of Sn , every 1-closure of an extension R ! A O is isomorphic to R, because A⌦n R⇠ = R. A⌦n Every degree-1 extension R ! A is isomorphic to R as an algebra of itself.
In the end, we will prove Theorem 1, which classifies G-closures of finite separable field extensions, and which grounds the theory of G-closures in classical Galois theory. We begin by analyzing the situation for trivial ´etale algebras, algebras of the form R ! Rn . We then use the fact that ´etale extensions are precisely those which become trivial after changing the base to an ´etale cover, to show that each G-closure of an ´etale extension is ´etale. Next, we employ an antiequivalence between the category of ´etale extensions and the category of finite sets equipped with a continuous action by a certain profinite group, in order to clasify G-closures of ´etale extensions.
In this chapter, we will always write the polynomial m(x) as m(x) = xn s1 x n 1 + s2 x n 2 . . + ( 1)n sn , so that in R[x]/(m(x)) we have sk = sk (x) for each k 2 [n]. and we will investigate G-closures of monogenic extensions for various groups G. First, we will discuss Gclosures of extensions with degree 0, 1, or 2, which are all (locally) monogenic. Then we tackle the case of intransitive G-closures, and show that Sk1 ⇥ . . ⇥ Skm -closures correspond to factorizations of m into factors of degree k1 , .