578 research outputs found

    Analysis of the classical cyclotomic approach to fermat's last theorem

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    We give again the proof of several classical results concerning the cyclotomic approach to Fermat's last theorem using exclusively class field theory (essentially the reflection theorems), without any calculations. The fact that this is possible suggests a part of the logical inefficiency of the historical investigations. We analyze the significance of the numerous computations of the literature, to show how they are probably too local to get any proof of the theorem. However we use the derivation method of Eichler as a prerequisite for our purpose, a method which is also local but more effective. Then we propose some modest ways of study in a more diophantine context using radicals; this point of view would require further nonalgebraic investigations.Comment: Publications Math\'ematiques UFR Sciences Techniques Besan\c{c}on 2010 (2010) 85-11

    A Cup Product in the Galois Cohomology of Number Fields

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    Let K be a number field containing the group of n-th roots of unity and S a set of primes of K including all those dividing n and all real archimedean places. We consider the cup product on the first Galois cohomology group of the maximal S-ramified extension of K with coefficients in n-th roots of unity, which yields a pairing on a subgroup of the multiplicative group of K containing the S-units. In this general situation, we determine a formula for the cup product of two elements which pair trivially at all local places. Our primary focus is the case that K is the cyclotomic field of p-th roots of unity for n = p an odd prime and S consists of the unique prime above p in K. We describe a formula for this cup product in the case that one element is a p-th root of unity. We explain a conjectural calculation of the restriction of the cup product to p-units for all p < 10,000 and conjecture its surjectivity for all p satisfying Vandiver's conjecture. We prove this for the smallest irregular prime p = 37, via a computation related to the Galois module structure of p-units in the unramified extension of K of degree p. We describe a number of applications: to a product map in K-theory, to the structure of S-class groups in Kummer extensions of K, to relations in the Galois group of the maximal pro-p extension of K unramified outside p, to relations in the graded Z_p-Lie algebra associated to the representation of the absolute Galois group of Q in the outer automorphism group of the pro-p fundamental group of P^1 minus three points, and to Greenberg's pseudo-nullity conjecture.Comment: final versio

    Ideal class groups of cyclotomic number fields II

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    We first study some families of maximal real subfields of cyclotomic fields with even class number, and then explore the implications of large plus class numbers of cyclotomic fields. We also discuss capitulation of the minus part and the behaviour of p-class groups in cyclic ramified p-extensions

    Vanishing of eigenspaces and cyclotomic fields

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    We use a result of Thaine to give an alternative proof of the fact that, for a prime p>3 congruent to 3 modulo 4, the component e_{(p+1)/2} of the p-Sylow subgroup of the ideal class group of \mathbb Q(\zeta_{p}) is trivial.Comment: 6 pages, minor corrections made, to appear in the International Mathematics Research Notice

    On the failure of pseudo-nullity of Iwasawa modules

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    We consider the family of CM-fields which are pro-p p-adic Lie extensions of number fields of dimension at least two, which contain the cyclotomic Z_p-extension, and which are ramified at only finitely many primes. We show that the Galois groups of the maximal unramified abelian pro-p extensions of these fields are not always pseudo-null as Iwasawa modules for the Iwasawa algebras of the given p-adic Lie groups. The proof uses Kida's formula for the growth of lambda-invariants in cyclotomic Z_p-extensions of CM-fields. In fact, we give a new proof of Kida's formula which includes a slight weakening of the usual assumption that mu is trivial. This proof uses certain exact sequences involving Iwasawa modules in procyclic extensions. These sequences are derived in an appendix by the second author.Comment: 26 page
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