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Comparing orders of Selmer groups. (English) Zbl 1098.11056

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 \(|S(F/K)|/|G(F/K)= |U_{K,n}| n^{r_1+r_2}/\prod_{v\in P(K)} e_v\).

MSC:

11R37 Class field theory
11R20 Other abelian and metabelian extensions
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References:

[1] Henri Cohen, Francisco Diaz y Diaz, Michel Olivier, Counting Cyclic Quartic Extensions of a Number Field. Journal de Théorie des Nombres de Bordeaux 17 (2005), 475-510. · Zbl 1090.11068
[2] Georges Gras, Class Field Theory, from theory to practice. Springer Monographs in Mathematics, second edition 2005. · Zbl 1019.11032
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