scholarly journals On the least prime ideal and Siegel zeros

2016 ◽  
Vol 12 (08) ◽  
pp. 2201-2229 ◽  
Author(s):  
Asif Zaman
Keyword(s):  
Prime Ideal ◽  
Number Field ◽  
Quadratic Forms ◽  
Class Group ◽  
Binary Quadratic Forms ◽  
Integral Ideal ◽  
Siegel Zero ◽  
Siegel Zeros ◽  
Special Case ◽  
Exceptional Character

Let [Formula: see text] be a number field, [Formula: see text] be an integral ideal, and [Formula: see text] be the associated narrow ray class group. Suppose [Formula: see text] possesses a real exceptional character [Formula: see text], possibly principal, with a Siegel zero [Formula: see text]. For [Formula: see text] satisfying [Formula: see text] [Formula: see text], we establish an effective [Formula: see text]-uniform Linnik-type bound with explicit exponents for the least norm of a prime ideal [Formula: see text]. A special case of this result is a bound for the least rational prime represented by certain binary quadratic forms.

On the Integral Extensions of Quadratic Forms Over Local Fields

1970 ◽  
Vol 22 (2) ◽  
pp. 297-307 ◽  
Author(s):  
Melvin Band
Keyword(s):  
Prime Ideal ◽  
Local Field ◽  
Quadratic Forms ◽  
Sufficient Conditions ◽  
Local Fields ◽  
Ring Of Integers ◽  
Finite Dimensional ◽  
Integral Analogue ◽  
Necessary And Sufficient ◽  
Special Case

Let F be a local field with ring of integers and unique prime ideal (p). Suppose that V a finite-dimensional regular quadratic space over F, W and W′ are two isometric subspaces of V (i.e. τ: W → W′ is an isometry from W to W′). By the well-known Witt's Theorem, τ can always be extended to an isometry σ ∈ O(V).The integral analogue of this theorem has been solved over non-dyadic local fields by James and Rosenzweig [2], over the 2-adic fields by Trojan [4], and partially over the dyadics by Hsia [1], all for the special case that W is a line. In this paper we give necessary and sufficient conditions that two arbitrary dimensional subspaces W and W′ are integrally equivalent over non-dyadic local fields.

scholarly journals Higher genus theory

2020 ◽  
Author(s):  
Peter Koymans ◽  
Carlo Pagano
Keyword(s):  
Number Field ◽  
Quadratic Forms ◽  
Class Group ◽  
Quadratic Extension ◽  
Number Fields ◽  
Explicit Description ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Genus Theory ◽  
Narrow Class

Abstract In $1801$, Gauss found an explicit description, in the language of binary quadratic forms, for the $2$-torsion of the narrow class group and dual narrow class group of a quadratic number field. This is now known as Gauss’s genus theory. In this paper, we extend Gauss’s work to the setting of multi-quadratic number fields. To this end, we introduce and parametrize the categories of expansion groups and expansion Lie algebras, giving an explicit description for the universal objects of these categories. This description is inspired by the ideas of Smith [ 16] in his recent breakthrough on Goldfeld’s conjecture and the Cohen–Lenstra conjectures. Our main result shows that the maximal unramified multi-quadratic extension $L$ of a multi-quadratic number field $K$ can be reconstructed from the set of generalized governing expansions supported in the set of primes that ramify in $K$. This provides a recursive description for the group $\textrm{Gal}(L/\mathbb{Q})$ and a systematic procedure to construct the field $L$. A special case of our main result gives an upper bound for the size of $\textrm{Cl}^{+}(K)[2]$.

scholarly journals On some class number relations for Galois extensions

1987 ◽  
Vol 107 ◽  
pp. 121-133 ◽  
Author(s):  
Takashi Ono
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Class Number ◽  
Quadratic Forms ◽  
Algebraic Number Field ◽  
Narrow Sense ◽  
Binary Quadratic Forms ◽  
Finite Degree ◽  
Algebraic Tori ◽  
Number Of Classes

Let k be an algebraic number field of finite degree over Q, the field of rationals, and K be an extension of finite degree over k. By the use of the class number of algebraic tori, we can introduce an arithmetical invariant E(K/k) for the extension K/k. When k = Q and K is quadratic over Q, the formula of Gauss on the genera of binary quadratic forms, i.e. the formula where = the class number of K in the narrow sense, the number of classes is a genus of the norm form of K/Q and tK = the number of distinct prime factors of the discriminant of K, may be considered as an equality between E(K/Q) and other arithmetical invariants of K.

scholarly journals Composition of binary quadratic forms over number fields

2021 ◽  
Vol 71 (6) ◽  
pp. 1339-1360
Author(s):  
Kristýna Zemková
Keyword(s):  
Number Field ◽  
Quadratic Forms ◽  
Number Fields ◽  
Binary Quadratic Form ◽  
Base Number ◽  
Binary Quadratic Forms ◽  
Quadratic Number Field ◽  
Totally Positive ◽  
Equivalent Conditions ◽  
Narrow Class

Abstract In this article, the standard correspondence between the ideal class group of a quadratic number field and the equivalence classes of binary quadratic forms of given discriminant is generalized to any base number field of narrow class number one. The article contains an explicit description of the correspondence. In the case of totally negative discriminants, equivalent conditions are given for a binary quadratic form to be totally positive definite.

THE CUBIC CONGRUENCE x3 + Ax2 + Bx + C ≡ 0 (mod p) AND BINARY QUADRATIC FORMS II

2001 ◽  
Vol 64 (2) ◽  
pp. 273-274 ◽  
Author(s):  
BLAIR K. SPEARMAN ◽  
KENNETH S. WILLIAMS
Keyword(s):  
Quadratic Forms ◽  
Class Group ◽  
Binary Quadratic Forms ◽  
Form Class ◽  
Mod P

It is shown that the splitting modulo a prime p of a given monic, integral, irreducible cubic with non-square discriminant is equivalent to p being represented by forms in a certain subgroup of index 3 in the form class group of discriminant equal to the discriminant of the field defined by the cubic.

scholarly journals An extended inhomogeneous minimum

1976 ◽  
Vol 22 (4) ◽  
pp. 431-441
Author(s):  
E. S. Barnes
Keyword(s):  
Number Field ◽  
Class Number ◽  
Quadratic Forms ◽  
Quadratic Number ◽  
Binary Quadratic Forms ◽  
Quadratic Number Field ◽  
Inhomogeneous Minimum ◽  
Norm Form

AbstractA new arithmetic invariant E(f) is defined for integral binary quadratic forms f. It has the property that, denoting by fm the norm-form of a quadratic number field Q(m), E(fm) < 1 if and only if Q(m) has class number one.

scholarly journals Václav Šimerka: quadratic forms and factorization

2013 ◽  
Vol 16 ◽  
pp. 118-129
Author(s):  
F. Lemmermeyer
Keyword(s):  
Quadratic Forms ◽  
Class Group ◽  
Binary Quadratic Forms ◽  
Carmichael Numbers

AbstractIn this article we show that the Czech mathematician Václav Šimerka discovered the factorization of $\frac{1}{9} (1{0}^{17} - 1)$ using a method based on the class group of binary quadratic forms more than 120 years before Shanks and Schnorr developed similar algorithms. Šimerka also gave the first examples of what later became known as Carmichael numbers.

REPRESENTATION OF PRIMES BY THE PRINCIPAL FORM OF NEGATIVE DISCRIMINANT $\Delta$ WHEN $h(\Delta)$ IS 4

1994 ◽  
Vol 25 (4) ◽  
pp. 321-334
Author(s):  
KENNETH S. WILLIAMS ◽  
D. LIU
Keyword(s):  
Cyclic Group ◽  
Quadratic Forms ◽  
Class Group ◽  
Positive Definite ◽  
Negative Integer ◽  
Binary Quadratic Forms ◽  
Form Class ◽  
Quartic Polynomial ◽  
Mod P

Let $\Delta$ be a negative integer which is congruent to 0 or 1 (mod 4). Let $H(\Delta)$ denote the form class group of classes of positive-definite, primitive integral binary quadratic forms $ax^2 +bxy +cy^2$ of discriminant $\Delta$. If $H(\Delta)$ is a cyclic group of order 4, an explicit quartic polynomial $\rho \Delta(x)$ of the form $x^4-bx^2 +d$ with integral coefficients is determined such that for an odd prime $p$ not dividing $\Delta$, $p$ is represented by the principal form of discriminant $\Delta$ if and only if the congruence $\rho \Delta(x) \equiv 0$ (mod $p$) has four solutions.

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