Comparing orders of Selmer groups
The author calculates the quotient of the orders of two Selmer groups, using two different methods based upon class field theory, Galois groups or Kummer radicals of some particular extensions, the ambiguous class number formula, and local criteria. In particular, let \(K^*\) be the multiplicative group of a number field \(K\), \(I_K\) the group of fractional ideals of \(K\), \((x)\) the principal fractional ideal generated by \(x\), with \(x\in K^*\), and \(P_K\subseteq I_K\) the group of principal fractional ideals of \(K\). If \(F\) is an algebraic extension of \(K\), let \(N= N_{F/K}\) denote the norm map from \(F\) to \(K\). For a given cyclic extension of number fields \(F/K\) with degree \(n\), the author defines the groups \(S(F/K)= \{x\in K^*\mid(x)\in \{I^n_K, x\in N(F^*)\}/K^*\), and \(G(F/K)= \}I\in I_K\mid\exists J\in I_F, N(J)= I\}/I^n_K N(P_F)\). Let \(U_{K,n}\) denote the set of \(n\)th roots of unity in \(K\), \(P1(K)\) the set of places of \(K\), \(r_1\) the number of real places of \(K\), and \(r_2\) the number of complex places of \(K\). For a given finite place \(v\) of \(K\), let \(e_v\) denote the ramification index of \(v\) in \(F/K\). When \(v\) is an infinite place of \(K\), define \(e_v= 1\) when \(v\) totally splits at \(v\), and \(e_v= 2\) in the other case; \(v\) is viewed as a ramified place if \(e_v= 2\). The author's main result establishes the formula that if \(F/K\) is a cyclic extension of number fields with degree \(n\), then \(|S(F/K)|/|G(F/K)= |U_{K,n}| n^{r_1+r_2}/\prod_{v\in P(K)} e_v\).
scientific article; zbMATH DE number 5016569