منابع مشابه
Cyclotomic Fields
Cyclotomic fields are an interesting laboratory for algebraic number theory because they are connected to fundamental problems Fermat’s Last Theorem for example and also have relatively simple algebraic properties that makes them an excellent laboratory for results in algebraic number theory. I will assume that you are familiar with basic algebraic number theory. Namely, the unique factorizatio...
متن کاملClasses of Cyclotomic Fields
As one will see in n◦4, these two theorems are deduced from a structure theorem for a certain group of operators. This group is constructed thus: let Xn be the p-component of the ideal class group of Kn; it is a finite abelian p-group, of order p en , which is acted opon by the Galois group G(Kn/Q) and in particular its subgroup Γn = G(Kn/K0). In passing to the projective limit over n with the ...
متن کاملDiophantine Equations in Cyclotomic Fields
where p is a given rational prime? It is almost trivial (from the theory of the Gaussian sum or otherwise) that a solution exists with g =p; it is less trivial that a solution also exists when g = p+p+l; but it is not asserted that solutions do not exist for other values of g. While we are unable to give anything like a complete answer to the problem proposed, we can prove something in this dir...
متن کاملSparse Representation for Cyclotomic Fields
Currently, all major implementations of cyclotomic fields as well as number fields, are based on a dense model where elements are represented either as dense polynomials in the generator of the field or as coefficient vectors with respect to a fixed basis. While this representation allows for the asymptotically fastest arithmetic for general elements, it is unsuitable for fields of degree > 10 ...
متن کاملCyclotomic-intermediate Fields via Gauss Sums
Let p be an odd prime, and let m divide p−1. Let ζ = e and let ω = e. The field extension Q(ω) ⊂ Q(ω, ζ) is Galois with cyclic Galois group isomorphic to (Z/pZ)×. The unique field between Q(ω) and Q(ω, ζ) having degree m over Q(ω) takes the form Q(ω, τ) where τ is a Gauss sum, to be described below. Furthermore, under some conditions we can compute τ as an element α of Q(ω), thus expressing the...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1970
ISSN: 0002-9939
DOI: 10.1090/s0002-9939-1970-0257037-6