Geodesic
CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS
Glasgow mathematical journal, Tome 62 (2020) no. 2, pp. 323-353

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DOI

We 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. 
CAPRIGLIO, FILIPPO A. E. NUCCIO MORTARINO MAJNO DI. CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS. Glasgow mathematical journal, Tome 62 (2020) no. 2, pp. 323-353. doi: 10.1017/S0017089519000144
@article{10_1017_S0017089519000144,
     author = {CAPRIGLIO, FILIPPO A. E. NUCCIO MORTARINO MAJNO DI},
     title = {CLASS {NUMBER} {FORMULA} {FOR} {DIHEDRAL} {EXTENSIONS}},
     journal = {Glasgow mathematical journal},
     pages = {323--353},
     year = {2020},
     volume = {62},
     number = {2},
     doi = {10.1017/S0017089519000144},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089519000144/}
}
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