scholarly journals Use of the rabinowitsch polynomial to determine the class groups of a real quadratic field

1994 ◽  
Vol 50 (3) ◽  
pp. 435-443
Author(s):  
R.A. Mollin
Keyword(s):  
Class Group ◽  
Quadratic Field ◽  
Sufficient Condition ◽  
Real Quadratic Field ◽  
Necessary And Sufficient Condition ◽  
Class Groups ◽  
Necessary And Sufficient ◽  
Real Quadratic

The main result is a necessary and sufficient condition for the class group of a real quadratic field to be determined by primality properties of the well-known Rabinowitsch polynomial.

On Central Ω-Krull Rings and their Class Groups

1984 ◽  
Vol 36 (2) ◽  
pp. 206-239 ◽  
Author(s):  
E. Jespers ◽  
P. Wauters
Keyword(s):  
Polynomial Ring ◽  
Class Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Semi Group ◽  
Class Groups ◽  
Krull Domain ◽  
Direct Limits ◽  
Commutative Theory ◽  
Necessary And Sufficient

The aim of this note is to study the class group of a central Ω-Krull ring and to determine in some cases whether a twisted (semi) group ring is a central Ω-Krull ring. In [8] we defined an Ω-Krull ring as a generalization of a commutative Krull domain. In the commutative theory, the class group plays an important role. In the second and third section, we generalize some results to the noncommutative case, in particular the relation between the class group of a central Ω-Krull ring and the class group of a localization. Some results are obtained in case the ring is graded. Theorem 3.2 establishes the relation between the class group and the graded class group. In particular, in the P.I. case we obtain that the class group is equal to the graded class group. As a consequence of a result on direct limits of Ω-Krull rings, we are able to derive a necessary and sufficient condition in order that a polynomial ring R[(Xi)i∊I] (I may be infinite) is a central Ω-Krull ring.

Prime Producing Quadratic Polynomials and Class-Numbers of Real Quadratic Fields

1990 ◽  
Vol 42 (2) ◽  
pp. 315-341 ◽  
Author(s):  
Stéphane Louboutin
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Quadratic Polynomials ◽  
Real Quadratic Fields ◽  
Consecutive Integers ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Real Quadratic

Frobenius-Rabinowitsch's theorem provides us with a necessary and sufficient condition for the class-number of a complex quadratic field with negative discriminant D to be one in terms of the primality of the values taken by the quadratic polynomial with discriminant Don consecutive integers (See [1], [7]). M. D. Hendy extended Frobenius-Rabinowitsch's result to a necessary and sufficient condition for the class-number of a complex quadratic field with discriminant D to be two in terms of the primality of the values taken by the quadratic polynomials and with discriminant D (see [2], [7]).

Characterizations of r-potent matrices

1984 ◽  
Vol 96 (2) ◽  
pp. 213-222 ◽  
Author(s):  
Joseph P. McCloskey
Keyword(s):  
Quadratic Form ◽  
Random Vector ◽  
Symmetric Matrix ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Real Quadratic

A matrix A is said to be tripotent whenever A3 = A. The study of tripotent matrices is of statistical interest since if the n × 1 real random vector X follows an N(0, I) distribution and A is a symmetric matrix then the real quadratic form X′AX is distributed as the difference of two independently distributed X2 variates if and only if A3 = A. In fact, a necessary and sufficient condition that A is tripotent is that there exist two idempotent matrices B and C such that A = B – C, and BC = 0. Using properties of diagonalizable matrices, we will prove several algebraic characterizations of r-potent matrices that extend the known results for tripotent matrices. Our first result will be to obtain an analogous decomposition for an arbitrary r-potent matrix.

scholarly journals ON THE IDEAL CLASS GROUP OF CERTAIN QUADRATIC FIELDS

2010 ◽  
Vol 52 (3) ◽  
pp. 575-581 ◽  
Author(s):  
YASUHIRO KISHI
Keyword(s):  
Class Number ◽  
Class Group ◽  
Quadratic Field ◽  
Quadratic Fields ◽  
Ideal Class ◽  
Real Quadratic Field ◽  
Ideal Class Group ◽  
The Real ◽  
Real Quadratic ◽  
The Ideal

AbstractLet n(≥ 3) be an odd integer. Let k:= $\Q(\sqrt{4-3^n})\)$ be the imaginary quadratic field and k′:= $\Q(\sqrt{-3(4-3^n)})\)$ the real quadratic field. In this paper, we prove that the class number of k is divisible by 3 unconditionally, and the class number of k′ is divisible by 3 if n(≥ 9) is divisible by 3. Moreover, we prove that the 3-rank of the ideal class group of k is at least 2 if n(≥ 9) is divisible by 3.

The group Gal(k3(2)|k) for k = ℚ(−3,d) of type (3,3)

2016 ◽  
Vol 12 (07) ◽  
pp. 1951-1986 ◽  
Author(s):  
Abdelmalek Azizi ◽  
Mohamed Talbi ◽  
Mohammed Talbi ◽  
Aïssa Derhem ◽  
Daniel C. Mayer
Keyword(s):  
Galois Group ◽  
Class Group ◽  
Quadratic Field ◽  
Galois Groups ◽  
Numerical Verification ◽  
Real Quadratic Field ◽  
Class Field ◽  
Type Formula ◽  
Biquadratic Fields ◽  
Real Quadratic

Let [Formula: see text] denote the discriminant of a real quadratic field. For all bicyclic biquadratic fields [Formula: see text], having a [Formula: see text]-class group of type [Formula: see text], the possibilities for the isomorphism type of the Galois group [Formula: see text] of the second Hilbert [Formula: see text]-class field [Formula: see text] of [Formula: see text] are determined. For each coclass graph [Formula: see text], [Formula: see text], in the sense of Eick, Leedham-Green, Newman and O’Brien, the roots [Formula: see text] of even branches of exactly one coclass tree and, in the case of even coclass [Formula: see text], additionally their siblings of depth [Formula: see text] and defect [Formula: see text], turn out to be admissible. The principalization type [Formula: see text] of [Formula: see text]-classes of [Formula: see text] in its four unramified cyclic cubic extensions [Formula: see text] is given by [Formula: see text] for [Formula: see text], and by [Formula: see text] for [Formula: see text]. The theory is underpinned by an extensive numerical verification for all [Formula: see text] fields [Formula: see text] with values of [Formula: see text] in the range [Formula: see text], which supports the assumption that all admissible vertices [Formula: see text] will actually be realized as Galois groups [Formula: see text] for certain fields [Formula: see text], asymptotically.

Notes on a theorem of Cochran

1952 ◽  
Vol 48 (3) ◽  
pp. 443-446 ◽  
Author(s):  
G. S. James
Keyword(s):  
Quadratic Forms ◽  
Degrees Of Freedom ◽  
Mathematical Statistics ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Linear Forms ◽  
Standard Normal ◽  
Real Linear ◽  
Necessary And Sufficient ◽  
Real Quadratic

1. General remarks. The theorem that has come to be known as Cochran's theorem in works on mathematical statistics was published in these Proceedings in 1934(1). If x1, …, xn are independently distributed standard normal deviates, and q1, …, qk are k real quadratic forms in the xi with ranks n1, …, nk respectively, and such that then Cochran's Theorem II states that a necessary and sufficient condition that q1 …, qk are independently distributed in χ2 forms with n1, …, nk degrees of freedom is that Σnj = n. The necessity of the condition is obvious. Cochxan proves its sufficiency by expressing each qj as a sum, involving nj squares of real linear forms in the xi; it follows easily that the coefficients ci are in fact + 1, and that the transformation is orthogonal. The theorem then follows immediately from the properties of orthogonal transformations in relation to independent normal deviates.

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