Yamashita, Hiroshi On the rank of the first radical layer of a \(p\)-class group of an algebraic number field. (English) Zbl 0998.11063 Nagoya Math. J. 156, 85-108 (1999). Let \(p\) be a prime number, \(M/k\) a finite Galois extension of algebraic number fields, \(G = \text{Gal}(M/k).\) Suppose that \(M\) contains a primitive \(p\)th root of unity and that the \(p\)-Sylow subgroup of \(G\) is normal. Let \(K\) be the fixed field of this \(p\)-Sylow subgroup and \(H = \text{Gal} (K/k).\) The \(p\)-class group \(C\) of \(M\) is a module over \({\mathbb Z}_{p}G,\) and the quotient \(C/JC,\) where \(J\) is the Jacobson radical of \(\mathbb Z_{p} G,\) is a module over \({\mathbb F}_{p} H.\) The author studies the multiplicity of a given irreducible representation appearing in \(C/JC.\) Using Kummer theory and genus theory, he proves a formula giving this multiplicity partially. He applies this formula to study a cyclotomic (resp. a CM) field \(M\) such that \(C^{-}\) (resp. \(C^{+})\) is \({\mathbb Z}_{p} G\)-cyclic (resp. trivial) for \(p \not= 2.\) Reviewer: T.Nguyen Quang Do (Besançon) MSC: 11R29 Class numbers, class groups, discriminants Keywords:cyclotomic fields; CM fields; multiplicity × Cite Format Result Cite Review PDF Full Text: DOI References: [1] DOI: 10.2307/1970932 · Zbl 0395.12014 · doi:10.2307/1970932 [2] Nagoya Math. J 71 pp 61– (1978) · Zbl 0362.12009 · doi:10.1017/S0027763000021632 [3] Nagoya Math. J 90 pp 137– (1983) · Zbl 0502.12007 · doi:10.1017/S0027763000020389 [4] J. reine angew. Math 199 pp 165– (1958) [5] DOI: 10.1007/BF02568324 · Zbl 0796.11048 · doi:10.1007/BF02568324 [6] J. reine angew. Math 311/312 pp 215– (1979) [7] DOI: 10.2307/2373625 · Zbl 0334.12013 · doi:10.2307/2373625 [8] Methods of representation theory with applications to finite groups and orders I (1981) [9] Cambrige studies in advanced math 30 (1991) [10] Part A and Part B, North-Holland math. studies I (1992) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.