Class number problem for imaginary cyclic number fields. (English) Zbl 0920.11077
Let \(N\) be an imaginary cyclic number field of degree \(2n\), \(n\) greater than or equal to 5, and \(n\) not a 2-power. The authors determine both all fields \(N\) with relative class numbers less than or equal to 4, and all fields \(N\) with class numbers equal to their genus class numbers. The main results can be stated as follows, where \(N\) is defined as above: 1) There are exactly 20 fields \(N\) with relative class numbers less than or equal to 4; their degrees are less than or equal to 22 and their conductors are less than or equal to 91. 2) There are exactly 15 fields \(N\) with class numbers equal to their genus class numbers; their degrees are less than or equal to 20, their conductors are less than or equal to 91, and their class numbers are less than or equal to 3.
In order to prove these results, the authors make use of the Dedekind zeta function, class number formulas, primitive Dirichlet characters of a given order, properties of CM-fields, conductors, and genus fields, and an impressive amount of computation to obtain class numbers, relative class numbers, genus class numbers, and conductors. Much of the theory and techniques is based upon the previous work of Louboutin, for whom 12 references are cited. A key proposition used to obtain the above results is the following, where \(\overline{h}\) denotes the relative class number of a CM-field or imaginary abelian number field \(K\), and \(h_F\) (respectively \(g_F)\) denotes the class number (respectively genus class number) of an abelian number field \(F\):
1) Let \(K\) and \(L\) be two CM-fields with \(K\) contained in \(L\) and such that \([L:K]\) is odd. Then \(\overline h_k\) divides \(\overline h_L\).
2) Let \(K\) be an imaginary abelian number field of degree \(2n\) with maximal real subfield \(K_+\). Let \(t\) denote the number of rational primes \(p\) such that all prime ideals of \(K_+\) above \(p\) are ramified in the quadratic extension \(K/K_+\). If \(h_k=g_k\), then \(\overline h_k=2^{t-1+ \varepsilon}\), where \(\varepsilon=0\) or 1 according as the narrow genus field of \(K_+\) is real or imaginary.
3) Let \(F\) be an abelian number field. If \(h_F=g_F\), then for any subfield \(K\) of \(F\) we also have \(h_k=g_k\).
In order to prove these results, the authors make use of the Dedekind zeta function, class number formulas, primitive Dirichlet characters of a given order, properties of CM-fields, conductors, and genus fields, and an impressive amount of computation to obtain class numbers, relative class numbers, genus class numbers, and conductors. Much of the theory and techniques is based upon the previous work of Louboutin, for whom 12 references are cited. A key proposition used to obtain the above results is the following, where \(\overline{h}\) denotes the relative class number of a CM-field or imaginary abelian number field \(K\), and \(h_F\) (respectively \(g_F)\) denotes the class number (respectively genus class number) of an abelian number field \(F\):
1) Let \(K\) and \(L\) be two CM-fields with \(K\) contained in \(L\) and such that \([L:K]\) is odd. Then \(\overline h_k\) divides \(\overline h_L\).
2) Let \(K\) be an imaginary abelian number field of degree \(2n\) with maximal real subfield \(K_+\). Let \(t\) denote the number of rational primes \(p\) such that all prime ideals of \(K_+\) above \(p\) are ramified in the quadratic extension \(K/K_+\). If \(h_k=g_k\), then \(\overline h_k=2^{t-1+ \varepsilon}\), where \(\varepsilon=0\) or 1 according as the narrow genus field of \(K_+\) is real or imaginary.
3) Let \(F\) be an abelian number field. If \(h_F=g_F\), then for any subfield \(K\) of \(F\) we also have \(h_k=g_k\).
Reviewer: E.Benjamin (Belfast / Maine)
Keywords:
imaginary cyclic number fields; genus class numbers; relative class numbers; conductors; CM-fieldsReferences:
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