Power Integral Bases in Composits of Number Fields

1998 ◽  
Vol 41 (2) ◽  
pp. 158-165 ◽  
Author(s):  
István Gaál
Keyword(s):  
Number Fields ◽  
Parametric Family ◽  
Prime Degree ◽  
Index Form ◽  
Form Equation ◽  
Integral Bases ◽  
Quadratic Extensions ◽  
Totally Real

AbstractIn the present paper we consider the problem of finding power integral bases in number fields which are composits of two subfields with coprime discriminants. Especially, we consider imaginary quadratic extensions of totally real cyclic number fields of prime degree. As an example we solve the index form equation completely in a two parametric family of fields of degree 10 of this type.

scholarly journals Power integral bases in a parametric family of totally real cyclic quintics

1997 ◽  
Vol 66 (220) ◽  
pp. 1689-1697 ◽  
Author(s):  
István Gaál ◽  
Michael Pohst
Keyword(s):  
Parametric Family ◽  
Integral Bases ◽  
Totally Real

scholarly journals Finding Normal Integral Bases of Cyclic Number Fields of Prime Degree

2000 ◽  
Vol 30 (2) ◽  
pp. 129-136
Author(s):  
Vincenzo Acciaro ◽  
Claus Fieker
Keyword(s):  
Number Fields ◽  
Prime Degree ◽  
Integral Bases

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.

Estimates for Kloosterman sums for totally real number fields

2001 ◽  
Vol 2001 (535) ◽  
Author(s):  
R. W. Bruggeman ◽  
R. J. Miatello ◽  
I. Pacharoni
Keyword(s):  
Real Number ◽  
Number Fields ◽  
Kloosterman Sums ◽  
Totally Real
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