Vol. 40, No. 1, 1972

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Γ-extensions of imaginary quadratic fields

Robert Gold

Vol. 40 (1972), No. 1, 83–88

Let p be an odd rational prime and E0 = 𝒬(√ −-m-) a quadratic imaginary number field. There is a unique Γ extension E of E0 for the prime p which is absolutely abelian. For each positive integer n there is a subfield En of E which is cyclic of degree pn over E0 and by Iwasawa the exponent of p in the class number of En is of the form μpn + λn + c for sufficiently large n. We here examine the analytic formula for the class number of En and in the case p = 3 give a simple condition implying that ff = 0. It follows easily from this condition that there are infinitely many imaginary quadratic fields which have Γ-extensions for the prime 3 with the invariants μ = 0 while λ 1.

Mathematical Subject Classification
Primary: 12A65
Received: 12 March 1970
Published: 1 January 1972
Robert Gold