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Kida’s theorem for a class of nonnormal extensions. (English) Zbl 0678.12005
Let E/F be an extension of \({\mathbb{Z}}_ p\)-fields of CM-type of degree p. The authors prove the following generalization of a theorem of Kida relating the Iwasawa invariants of E and F: If the normal closure L of E/F has a subfield K with \([L:K]=p\) and K/F normal then \(\mu^-_ K=0\) implies \(\mu^-_ F=\mu^-_ E=0\) and \(\lambda^-_ E=\lambda^- _ F+(p-1)(\lambda^-_ K+t-\delta)/[K:F]\). Here t is the number of primes, not dividing p, in \(K^+\) which ramify in \(L^+\) and split completely in K, where \(K^+, L^+\) denote the maximal real subfields of K and L. Also \(\delta =1\) or 0 according as K contains the p-th roots of unity or not.
The authors give two proofs of this result. The first is arithmetic- algebraic in nature and uses previous results of the authors from Acta Arith. 46, 243-255 (1986; Zbl 0603.12003). The second proof is analytic in nature and uses techniques of W. M. Sinnott [Compos. Math. 53, 3-17 (1984; Zbl 0545.12011)] which relate p-adic L-functions to Iwasawa invariants.
Reviewer: C.Parry

MSC:
11R18 Cyclotomic extensions
11R42 Zeta functions and \(L\)-functions of number fields
11S40 Zeta functions and \(L\)-functions
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