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Capitulation in unramified quadratic extensions of real quadratic number fields

Published online by Cambridge University Press:  18 May 2009

E. Benjamin ,
F. Sanborn  and
C. Snyder
E. Benjamin
Affiliation:
Department of Mathematics, Unity College Unity, Maine 04988, USA
F. Sanborn
Affiliation:
Department of Mathematics, University of Maine, Orono, Maine 04469-5752, USA
C. Snyder
Affiliation:
Department of Mathematics, University of Maine, Orono, Maine 04469-5752, USA
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Let k be an algebraic number field and Ck its ideal class group (in the wider sense). Suppose K is a finite extension of k. Then we say that an ideal class of k capitulates in K if this class is in the kernel of the homomorphism

induced by extension of ideals from k to K (See Section 2 below). In [4], Iwasawa gives examples of real quadratic number fields, with distinct primes Pi ≡ 1 (mod 4), for which all the ideal classes of the 2-class group, Ck,2 (the 2-Sylow subgroup of Ck), capitulate in an unramified quadratic extension of k. In these examples, Ck,2 is abelian of type (2,2), i.e. isomorphic to ℤ/2ℤ×ℤ/2ℤ and so all four ideal classes capitulate.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 36 , Issue 3 , September 1994 , pp. 385 - 392
Copyright
Copyright © Glasgow Mathematical Journal Trust 1994

References

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