×

Quadratic equations over finite fields and class numbers of real quadratic fields. (English) Zbl 0898.11043

The authors look at finite fields \(\mathbb{F}_p\), where \(p>2\) is prime, and derive explicit formulae for the number of points of certain quadratic hypersurfaces in \(\mathbb{F}^n_p\), where \(n\in\mathbb{N}\). Furthermore, they show that the class number of \(\mathbb{Q}(\sqrt p)\) for \(p\equiv 1\pmod 4\) can be expressed in terms of their formulae. They conclude the paper with a formula for the validity of the Ankeny-Chowla conjecture to hold for \(\mathbb{Q}(\sqrt p)\).
Reviewer: R.Mollin (Calgary)

MSC:

11R29 Class numbers, class groups, discriminants
05A19 Combinatorial identities, bijective combinatorics
11R11 Quadratic extensions
05E15 Combinatorial aspects of groups and algebras (MSC2010)
11E04 Quadratic forms over general fields
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Agoh T (1989) A note on unit and class number of real quadratic fields. Acta Math Sinica (NS)5: 281-288 · Zbl 0701.11045 · doi:10.1007/BF02107554
[2] Ankeny NC, Artin E, Chowla S (1952) The class number of real quadratic fields. Ann Math56: 479-493 · Zbl 0049.30605 · doi:10.2307/1969656
[3] Ankeny NC, Chowla S (1962) A further note on the class number of real quadratic fields. Acta Arith7: 271-272 · Zbl 0214.30802
[4] Lidl R, Niederreiter H (1983) Finite Fields. Reading, MA: Addison-Wesley · Zbl 0554.12010
[5] Le M-H (1994) The number of solutions of a certain quadratic congruence related to the class number of \(\mathbb{Q}(\sqrt p )\) . Proc Amer Math Soc117: 1-3 · doi:10.1090/S0002-9939-1993-1110547-7
[6] Le M-H (1994) Upper bounds for class number of real quadratic fields. Acta Arith68: 141-144 · Zbl 0816.11055
[7] Orlik P, Solomon L (1980) Unitary reflection groups and cohomology. Invent Math59: 77-94 · Zbl 0452.20050 · doi:10.1007/BF01390316
[8] Solomon L (1963) Invariants of finite reflection groups. Nagoya Math J22 57-64 · Zbl 0117.27104
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.