Let $K$ be the quadratic field $\mathbf{Q}(\sqrt{m})$ with
discirimant $d = pq$. Using Legendre's theorem on the solvability
of the equation $ax^2 + by^2 = z^2$, we give necessary and
sufficient conditions for the class number of $K$ in the narrow
sense to be divisible by 8. The approach recovers known criteria
but is simpler and can be extended to compute the sylow 2-subgroup
of the ideal class group of quadratic fields.
Publié le : 2004-12-14
Classification:
Legendre's theorem,
2-part,
Gauss genus theorem,
11R29,
11R11
@article{1116442503,
author = {Basilla, Julius M.},
title = {The quadratic fields with discriminant divisible by exactly two primes and with ``narrow'' class number divisible by 8},
journal = {Proc. Japan Acad. Ser. A Math. Sci.},
volume = {80},
number = {6},
year = {2004},
pages = { 187-190},
language = {en},
url = {http://dml.mathdoc.fr/item/1116442503}
}
Basilla, Julius M. The quadratic fields with discriminant divisible by exactly two primes and with ``narrow'' class number divisible by 8. Proc. Japan Acad. Ser. A Math. Sci., Tome 80 (2004) no. 6, pp. 187-190. http://gdmltest.u-ga.fr/item/1116442503/