91 research outputs found
On a normal integral basis problem over cyclotomic -extensions (Algebraic Number Theory and Related Topics)
No full text
Local Units Modulo Gauss Sums
No full textAbstractFor a prime numberpand a number fieldk, letk∞/kbe the cyclotomic Zp-extension. LetA∞be the projective limit of thep-part of the ideal class group of each intermediate field ofk∞/k. Whenkis totally real, it is conjectured thatA∞is finite, namely that the characteristic polynomial char(A∞) ofA∞as aΛ-module is 1. We give an interpretation of char(A∞) (and hence, of the conjecture) in terms ofp-adic behaviour of certain Gauss sums whenkis a real abelian field (satisfying some conditions). Whenk=Q(cos(2π/p)), similar results are already obtained by Coleman [3], Kaneko and the author [9]
On the Class Numbers of the Maximal Real Subfields of Cyclotomic Function Fields
No full textAbstractFor a prime numberl, leth+lbe the class number of the maximal real subfield of thel-th cyclotomic field. For each natural numberN, it is plausible but not yet proved that there exist infinitely many prime numberslwithh+l>N. We prove an analogous assertion for cyclotomic function fields
HILBERT-SPEISER NUMBER FIELDS AND STICKELBERGER IDEALS; THE CASE p = 2
No full textWe say that a number field F satisfies the condition (H′2m) when any abelian extension of exponent dividing 2m has a normal basis with respect to rings of 2-integers. We say that it satisfies (H′
2∞) when it satisfies (H′
2m) for all m. We give a condition for F to satisfy (H'2m), and show that the imaginary quadratic fields F = Q(√−1) and Q(√−2) satisfy the very strong condition (H′
2∞) if the conjecture that h+2m = 1 for all m is valid. Here, h+2m) is the class number of the maximal real abelian field of conductor 2m
On normal integral bases of unramified abelian p-extensions over a global function field of characteristic p
No full textAbstractLet K be an algebraic function field of one variable over a finite field of characteristic p, and S a finite non-empty set of prime divisors of K. As the ring of integers of K, we take the ring of elements of K integral outside S. We prove that for a finite abelian p-extension L/K, it has a relative normal integral basis (NIB) if and only if it is unramified outside S. We also give a generator of NIB in an explicit form
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