Kummer theory without roots of unity
Let \(R\) be a connected commutative ring with group of units \(R^*\), \(p\) an odd prime which is a unit in \(R\), \(\Gamma_n\) a group cyclic of order \(q=p^n\) with generator \(\tau_n\), \(G_n\) its dual with generator \(\sigma_n\). The authors classify the group \(H(R,\Gamma_n)\) of Galois extensions of \(R\) with group \(\Gamma_n\) having normal basis (c.f. \textit{S. U. Chase}, \textit{D. K. Harrison} and \textit{A. Rosenberg} [Mem. Am. Math. Soc. 52, 15--33 (1965; Zbl 0143.05902)]); when \(R\) contains a primitive \(q\)-th root of unity, it is well-known that the group \(H(R,\Gamma_n)\) is isomorphic to \(R^*/R^{*q}\) by classical Kummer theory. The authors do not assume \(R\) contains roots of unity, and rather than adjoining a \(q\)-th root of unity, using the classical theory and then descending (\textit{C. Greither}'s approach in his 1988 München Habilitationsschrift, written after this paper) they associate to a normal basis of a \(\Gamma_n\)-Galois extension an element of the group ring \(R[G_n]\), and then identify the resulting classes of elements. More precisely, they define a subgroup \(C_n(R)\) of \(R(G_n)^*\), define a homomorphism \(\phi\) from \(C_n(R)\) to \(H(R,\Gamma_n)\), and identify the kernel of \(\phi\). To accomplish this, the authors first show that if \(A\) is a \(\Gamma_n\)-Galois extension of \(R\), then \(X_0,\dots,X_{q-1}\) forms a normal basis of \(A\) iff \(X=\sum_{i=0}^{q-1}X_i\sigma_n^{-i}\) is a unit of \(A[G_n]\) and \(\tau_n(X)=\sigma_n(X)\). They call such an \(X\) in \(A[G_n]\) a normal basis of \(A\) over \(R\). A normal basis \(X\) of \(A\) over \(R\) can be found which is normalized: if \(v_k: A[G_n]\to A[G_n]\) is induced by sending \(\sigma\) to \(\sigma^k\), then \(X\) in \(A[G_n]^*\) is normalized if \(v_0(X)= \mathrm{tr}(X_0)=1\) and \(v_{-1}(X)X=1\). Let \(q\), \(w\) be the multiplicative maps \(A[G_n]^*\to A[G_n]^*\) defined by \(q(x)=x^{p+1}/v_{p+1}(x)\); \(w(x)=x^p/v_p(x)\). Then \(q\) and \(w\) commute. If \(X\) is a normalized normal basis of \(A\) over \(R\), then \(Q=q(X)\) is a normalized element of \(R[G_n]^*\), as is \(W=w(X)\), and \(w(Q)=q(W)\). The group \(C_n(R)\) is defined to be the group of normalized elements \(Q\) of \(R[G_n]^*\) for which there exists normalized \(W\) in \(R[G_n]\) with \(w(Q)=q(W)\). The map \(q\) which sends normal bases of \(\Gamma_n\)-Galois extensions of \(R\) to elements of \(C_n(R)\) is \(1-1\) on isomorphism classes of \(\Gamma_n\)-Galois extensions with normal basis, and \(A\) is the trivial Galois extension of \(R\) iff for any normal basis \(X\) of \(A\), \(q(X)=q(\alpha)\) for some normalized \(\alpha\) in \(R[G_n]\). The main work of the paper is to define the map \(\phi\) from \(C_n(R)\) to \(H(R,\Gamma_n)\): that is, given an element \(Q\) of \(C_n(R)\), to construct a \(\Gamma_n\)-Galois extension \(A\) of \(R\) and a normal basis \(X\) of \(A\) which is in the image of a certain map \(\gamma\), such that \(q(X)=Q\). -- The group \(C_n(R)\) is a subgroup of \(R[G_n]\), but the authors also construct a homomorphism \(g\) from \(R[G_n]^*\) to \(C_n(R)\) which, when composed with \(\phi\), yields a surjection to \(H(R,\Gamma_n)\); the kernel of \(\phi \circ g\) is not identified. In the final section of the paper the authors examine \(\mathbb Z_p\)-extensions, infinite chains \(R=R_0\subseteq R_1\subseteq \ldots\subseteq R_{\infty}=\cup R_n\) where \(R_n\) is a Galois extension of \(R\) with group \(\Gamma_n\). The authors show that \(R_{\infty}\) is connected iff \(R_1\) is connected, and that if each \(R_n\) has a normal basis over \(R\), then the normal bases for various \(n\) may be chosen in a coherent way with respect to the classification via \(C_n(R)\). These results have proved useful in the authors' subsequent study of \(\mathbb Z_p\)-extensions of number fields [C. R. Math. Sci., Soc. R. Can. 9, 309--314 (1987; Zbl 0649.12010)].
scientific article; zbMATH DE number 4040046
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