scholarly journals L-values and the Fitting ideal of the tame kernel for relative quadratic extensions

2008 ◽  
Vol 131 (4) ◽  
pp. 389-402 ◽  
Author(s):  
Jonathan W. Sands
Keyword(s):  
Quadratic Extensions ◽  
Fitting Ideal ◽  
Tame Kernel ◽  
Relative Quadratic Extensions

scholarly journals VALUES AT s = -1 OF L-FUNCTIONS FOR MULTI-QUADRATIC EXTENSIONS OF NUMBER FIELDS, AND THE FITTING IDEAL OF THE TAME KERNEL

2009 ◽  
Vol 05 (03) ◽  
pp. 383-405
Author(s):  
JONATHAN W. SANDS
Keyword(s):  
Galois Extension ◽  
Number Fields ◽  
Sufficient Condition ◽  
Tate Conjecture ◽  
Quadratic Extensions ◽  
Fitting Ideal ◽  
Tame Kernel ◽  
Finite Set ◽  
Stickelberger Ideal ◽  
Totally Real

Fix a Galois extension [Formula: see text] of totally real number fields such that the Galois group G has exponent 2. Let S be a finite set of primes of F containing the infinite primes and all those which ramify in [Formula: see text], let [Formula: see text] denote the primes of [Formula: see text] lying above those in S, and let [Formula: see text] denote the ring of [Formula: see text]-integers of [Formula: see text]. We then compare the Fitting ideal of [Formula: see text] as a ℤ[G]-module with a higher Stickelberger ideal. The two extend to the same ideal in the maximal order of ℚ[G], and hence in ℤ[1/2][G]. Results in ℤ[G] are obtained under the assumption of the Birch–Tate conjecture, especially for biquadratic extensions, where we compute the index of the higher Stickelberger ideal. We find a sufficient condition for the Fitting ideal to contain the higher Stickelberger ideal in the case where [Formula: see text] is a biquadratic extension of F containing the first layer of the cyclotomic ℤ2-extension of F, and describe a class of biquadratic extensions of F = ℚ that satisfy this condition.

A Sequence of Results on Class Number Congruences

1993 ◽  
Vol 36 (2) ◽  
pp. 139-143
Author(s):  
Antone Costa
Keyword(s):  
Class Number ◽  
Positive Density ◽  
Quadratic Extensions ◽  
Totally Real ◽  
Relative Quadratic Extensions

AbstractLet p ≡ 1 mod 8 be a rational prime and let h(—p) be the class number of . In [1], Barrucand and Cohn show that h(-p) = 0 mod 8 iff p = x2 + 32y2 for some x,y € Z. In this article, we generalize their result to a family of relative quadratic extensions K/F, where Fk is the maximum totally real subfield of Q(ζ2k+2 ), and a power of a prime of Fk from a family of positive density.

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