Stickelberger's criterion, Galois algebras, and tame ramification in algebraic number fields
Let F be an algebraic number field in which the odd prime p is unramified. For a cyclic Galois extension K of F of degree p, let \({\mathcal S}_{K/F}\) be the set of primes of F which ramify tamely in K. The main theorem of this paper, generalizing work of \textit{S. Ullom} [J. Number Theory 6, 32-49 (1974; Zbl 0278.12004)], is a characterization of finite sets \({\mathcal S}=\{q_ 1,...,q_ m\}\) of prime ideals of F such that \({\mathcal S}={\mathcal S}_{K/F}\) for some K. The characterization involves the known necessary condition that each prime q in \({\mathcal S}\) splits completely in F(\(\zeta)\), \(\zeta\) a primitive pth root of unity. If \(Q_ i\) is a prime divisor of \(q_ i\) in F(\(\zeta)\), and \({\mathcal A}=\prod^{m}_{i=1}Q_ i\), then the characterization \({\mathcal S}={\mathcal S}_{K/F}\) is given in terms of a congruence condition on the images cl \({\mathcal A}^{\tau (m)}\) of the ideal class of \({\mathcal A}\) under the action of Stickelberger elements \(\tau\) (m) in \({\mathbb{Q}}\Delta\), \(\Delta =Gal({\mathbb{Q}}(\zeta)/{\mathbb{Q}})\). The author characterizes in a similar way those \({\mathcal S}\) with \({\mathcal S}={\mathcal S}_{K/F}\) for K tamely ramified or with normal integral basis. To obtain these results, the author first gives an arithmetic classification of the set of Galois R-algebras with Galois group G, cyclic of prime order p, when R is a commutative ring with 2p a non-zero divisor. Here a Galois R-algebra A is in the sense of Hasse: \(A=RGa\cong RG\) as RG-module. The classification is obtained by letting S be the cyclotomic extension obtained by adjoining a formal pth root of unity to R, computing the resolvant (a\(| \chi)\) in \(A\otimes S\) for \(\chi\) a generator of the character group (into S) of G, and associating to A the element \(\theta =(a| \chi)^ p/tr(a)\) in \(A\otimes S\). The resolvant powers \(\theta\) arising in this way are characterized by congruence conditions involving the images of \(\theta\) under the action of elements of the \({\mathbb{Z}}\)-submodule \({\mathcal B}\) of \({\mathbb{Q}}\Delta\) such that \(\tau\) (p) \({\mathcal B}\subseteq {\mathbb{Z}}\Delta\), together with an integrality condition which in interesting number-theoretic cases is implied by the Stickelberger conditions. In the unramified case, that is, when A is a Galois extension of R with normal basis, the map \(A\mapsto \theta\) induces an isomorphism from the Harrison group of Galois extensions of R with normal basis, to a factor group of the group \(U(S_{\pi^ p})^{\tau (p)}\), the images under \(\tau\) (p) of units \(\theta\) of S congruent to 1 modulo \(\pi\) \({}^ p\), \(\tau =1-\zeta\). This extends to rings R without pth roots of unity a result of the reviewer [Proc. Lond. Math. Soc., III. Ser. 35, 407-422 (1977; Zbl 0374.13002)].
scientific article; zbMATH DE number 3889660