Residuacity properties of real quadratic units
TLDR
In this article, necessary and sufficient conditions for representing certain classes of primes by given quadratic forms are found by generalizing techniques of rational number theory, and the main result is that if m = 5 or 13, and if p is a rational prime such that ( − 1 p ) = 1 = ( m p ), then a necessary and necessary condition that x 2 + 4 my 2 = p for some rational integers x and y is that [ ϵ m p ] = 1, where ϵm denotes the fundamental unit of the field Q(m 1About:
This article is published in Journal of Number Theory.The article was published on 1973-08-01 and is currently open access. It has received 16 citations till now. The article focuses on the topics: Quadratic field & Legendre symbol.read more
Citations
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Journal ArticleDOI
Cubic residues and binary quadratic forms
TL;DR: In this article, a general criterion for m to be a quartic residue (mod p ) in terms of appropriate binary quadratic forms was given, where m ∈ Z and p ∤ m is an odd prime.
Journal ArticleDOI
Fundamental units of real quadratic fields of odd class number
Zhe Zhang,Qin Yue +1 more
TL;DR: In this paper, a real quadratic field with odd class number and its fundamental unit ϵ d = x + y d > 1 satisfies a congruence relation about x, y explicitly.
Journal Article
On the quartic character of quadratic units.
TL;DR: In this paper, the authors considered the question of whether a fundamental unit q in the quadratic field Q (j/g) is a residue of a prime p of which g is a quadratically residue.
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On the number of incongruent residues of x4+ax2+bx modulo p
TL;DR: In this paper, the authors mainly determine the number V p (x 4 + a x 2 + b x ) of incongruent residues of the modulo p and reveal the connections with elliptic curves over the field F p of p elements.
Journal ArticleDOI
On the quadratic character of quadratic units
TL;DR: In this paper, the congruences for quadratic residue (mod p) were established for the Lucas sequence, where U n is a Lucas sequence defined by U 0 = 0, U 1 = 1 and U n + 1 = b U n+k 2 U n − 1 ( n ⩾ 1 ).
References
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Book ChapterDOI
Die Theorie der algebraischen Zahlkörper
TL;DR: In this article, a rationalen Funktionen of α, β,..., ϰ with rationalen Zahlenkoeffizienten ein in sich abgeschlossenes System von algebraischen Zahlen, welches Zahlkorper, Korper (R. Dedekind) oder Rationalitatsbereich (L. Kronecker) genannt wird.
Journal Article
Note on primes of type x2 + 32y2, class number, and residuacity.
H. Cohn,P. Barrucand +1 more
Book ChapterDOI
Über den Dirichletschen biquadratischen Zahlkörper
TL;DR: The Dirichletsche Zahlkorper genannt Dirichlet as discussed by the authors is a biquadratische ZahlKorper, in which die imaginare Einheit i and mithin all jene Gausschen imaginaren Zahlen enthalt.