Generalized invariant classes
Let \(K/k\) be a cyclic extension of number fields, with Galois group \(G\), and let \(L\) be a finite abelian extension of \(K\), Galois over \(k\). We give an explicit formula for \(\# \text{Gal} (L/K)^ G\), which is also the degree over \(K\) of the maximal subfield of \(L\), abelian over \(k\). This formula generalizes that of the author [Ann. Inst. Fourier 23, No. 3, 1-48 (1973; Zbl 0276.12013); ibid. 23 (1973), No. 4, 1-44 (1974; Zbl 0269.12004)], of \textit{A. Harnchoowong} and \textit{W.-C. W. Li} [J. Number Theory 36, 354-372 (1990; Zbl 0723.11053)], and \textit{J.-F. Jaulent} (Thèse) [Publ. Math. Fac. Sci. Besançon, Théor. Nombres 1984/85- 1985/86 (1986; Zbl 0614.12006)]. A standard application of such a formula is the determination of the structure of a generalized \(p\)-class group \(Cl_{K,{\mathfrak m}}\) in a \(p\)- cyclic extension \(K/k\), by means of the computation of the filtration \((M_ i )_{i\geq 0}\) such that \(M_{i+1}/ M_ i= (C\ell_{K,{\mathfrak m}}/ M_ i)^ G\) and \(M_ 0=1\). Our approach differs from classical ones in the sense it appears as a ``translation'' of any classical invariant class formula (e.g. \(\# C\ell_ K\) (C. Chevalley, Y. Furuta)) by means of a convenient exact sequence.
scientific article; zbMATH DE number 757988