scholarly journals The Brauer–Kuroda formula for higher S-class numbers in dihedral extensions of number fields

2012 ◽  
Vol 151 (3) ◽  
pp. 217-239 ◽  
Author(s):  
Luca Caputo
Keyword(s):  
Number Fields ◽  
Class Numbers ◽  
Dihedral Extensions

scholarly journals CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS

2019 ◽  
Vol 62 (2) ◽  
pp. 323-353
Author(s):  
LUCA CAPUTO ◽  
FILIPPO A. E. NUCCIO MORTARINO MAJNO DI CAPRIGLIO
Keyword(s):  
Integral Representation ◽  
Class Number ◽  
Number Fields ◽  
Class Numbers ◽  
Algebraic Proof ◽  
Arithmetic Means ◽  
Class Number Formula ◽  
Number Formula ◽  
Explicit Bounds ◽  
Dihedral Extensions

AbstractWe give an algebraic proof of a class number formula for dihedral extensions of number fields of degree 2q, where q is any odd integer. Our formula expresses the ratio of class numbers as a ratio of orders of cohomology groups of units and allows one to recover similar formulas which have appeared in the literature. As a corollary of our main result, we obtain explicit bounds on the (finitely many) possible values which can occur as ratio of class numbers in dihedral extensions. Such bounds are obtained by arithmetic means, without resorting to deep integral representation theory.

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