Abstract
We show how the decomposition of primes in certain dihedral extensions L of the rationals enables us to obtain results concerning representations of powers of primes by binary quadratic forms and treat here in detail the case of\(L = Q\left( { \sqrt {\varepsilon _m } ,\sqrt { - \varepsilon _m } } \right)\), where m is a square free positive integer such that the norm of the fundamental unit εm of\(Q\left( {\sqrt m } \right)\) is −1. Other cases will be treated in subsequent papers.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D.A. BUELL, P.A. LEONARD and K.S. WILLIAMS, Note on the quadratic character of a quadratic unit, submitted for publication
C.F. GAUSS, Disquisitiones Arithmeticae, translated into English by Arthur A. Clarke, Yale University Press, 1966
F. HALTER-KOCH, Arithmetische Theorie der Normalkörper von 2. Potenzgrad mit Dieder gruppe, J. Number Theory, 3 (1971), pp. 412–443
D. HILBERT, Über die Theorie des relativquadratischen Zahlkörpers, Math. Ann. 51 (1899), 1–127
P. KAPLAN, Sur le 2-groupe des classes d'idéaux des corps quadratiques, J. Reine Angew. Math 283/284 (1976), 313–363
P. KAPLAN, Cours d'Arithmétique, UER de Mathématiques, Université de Nancy I, (1973)
K.S. WILLIAMS, On the evaluation of (\(\varepsilon _{q_1 q_2 } /p\)), Rocky Moutain J. Math. (to appear)
Author information
Authors and Affiliations
Additional information
Research supported by Natural Sciences and Engineering Research Council Canada Grant No A-7233
Rights and permissions
About this article
Cite this article
Kaplan, P., Williams, K.S. An Artin character and representations of primes by binary quadratic forms. Manuscripta Math 33, 339–356 (1981). https://doi.org/10.1007/BF01798232
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01798232