A remark on the capitulation problem
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A remark on the capitulation problem Ali Mouhib1
© Akadémiai Kiadó, Budapest, Hungary 2016
Abstract We determine explicitly an infinite family of imaginary cyclic number fields k, such that the 2-class group of k is elementary with arbitrary large 2-rank and capitulates in an unramified quadratic extension K . The infinitely many number fields k and K have the same Hilbert 2-class field and an infinite Hilbert 2-class field tower. Keywords Class group · Unit group · Capitulation problem · ZZ2 -extension Mathematics Subject Classification 11R29 · 11R32 · 11R37 · 11R23
1 Introduction Let F /F be an arbitrary finite extension of number fields, C F (resp. C F ) the class group of F (resp. F ) and F 1 the Hilbert class field of F. Let the j denote the basic homomorphism j : C F −→ C F induced by extension of ideals from F to F . We say that an ideal class c of F capitulates in F if c ∈ ker ( j). Also, we say that C F capitulates in F if each ideal class of F capitulates in F . The principal ideal theorem, one of the most famous results on the capitulation problem, states that each ideal class of F capitulates in F 1 . This result was conjectured by D. Hilbert, reduced to a problem on finite groups by E. Artin and finally proved by P. Furtwängler around 1930. Let F be a proper subextension of F 1 /F. One of the interesting problems is to study the capitulation in the extension F /F, more precisely determining the number of elements of C F which capitulate in F . Let p be a prime number, C p,k the p-class group of k and F p1 the Hilbert p-class field of F. Determining a proper subextension of F 1 /F in which C F
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Ali Mouhib [email protected] LSI, Sciences and Engineering Laboratory, Polydisciplinary Faculty of Taza, Université Mohammed Ben Abdellah, Fes, B.P 1223, Taza-Gare, Morocco
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capitulates is reduced to the existence of a prime number p and a proper subextension of F p1 /F in which C p,F capitulates. In [5], K. Iwasawa determines an infinite family of quadratic number fields F with C2,F isomorphic to the group ZZ/2ZZ × ZZ/2ZZ and C2,F capitulating in a quadratic unramified extension of F. In this article, using some results of Iwasawa theory of ZZ2 -extensions, we determine an infinite family of number field extensions F /F, such that F /F is a quadratic unramified extension, F/Q is a cyclic 2-extension with C2,F of arbitrary large 2-rank, F and F have the same Hilbert 2-class field and C2,F capitulates in F . Let √and be distinct prime numbers such that ≡ −1 (mod 4), ≡ 3 (mod 8) and k = Q( − ). Let Q∞ be the cyclotomic ZZ2 -extension of Q and for each positive integer n, let Qn be the n-th layer of Q∞ . Also, let k∞ be the cyclotomic ZZ2 -extension of k and for each nonnegative integer n, let kn denote the n-th layer of k∞ , so kn = Qn k. Denote by Mn the proper intermediate number field of kn+1 /Qn other than Qn+1 and kn . Denote by m the integer such that m = v2 ( + 1) − 2, so it is clear that Qm is the decomposition field of in Q∞ /Q. The mai
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