ON STARK’S CLASS NUMBER CONJECTURE AND THE GENERALISED BRAUER–SIEGEL CONJECTURE

2022 ◽  
pp. 1-13
Author(s):  
PENG-JIE WONG
Keyword(s):  
Class Number ◽  
Number Fields ◽  
Galois Closure ◽  
Stark's Conjecture

Abstract Stark conjectured that for any $h\in \Bbb {N}$ , there are only finitely many CM-fields with class number h. Let $\mathcal {C}$ be the class of number fields L for which L has an almost normal subfield K such that $L/K$ has solvable Galois closure. We prove Stark’s conjecture for $L\in \mathcal {C}$ of degree greater than or equal to 6. Moreover, we show that the generalised Brauer–Siegel conjecture is true for asymptotically good towers of number fields $L\in \mathcal {C}$ and asymptotically bad families of $L\in \mathcal {C}$ .

scholarly journals On S-class number relations of algebraic tori in Galois extensions of global fields

1991 ◽  
Vol 124 ◽  
pp. 133-144 ◽  
Author(s):  
Masanori Morishita
Keyword(s):  
Class Number ◽  
Quadratic Forms ◽  
Euler Number ◽  
Number Fields ◽  
Binary Quadratic Forms ◽  
Algebraic Number Fields ◽  
Algebraic Tori ◽  
Genus Theory ◽  
Class Number Formula ◽  
Number Formula

As an interpretation and a generalization of Gauss’ genus theory on binary quadratic forms in the language of arithmetic of algebraic tori, Ono [02] established an equality between a kind of “Euler number E(K/k)” for a finite Galois extension K/k of algebraic number fields and other arithmetical invariants associated to K/k. His proof depended on his Tamagawa number formula [01] and Shyr’s formula [Sh] which follows from the analytic class number formula of a torus. Later, two direct proofs were given by Katayama [K] and Sasaki [Sa].

scholarly journals A Density of Ramified Primes

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Stephanie Chan ◽  
Christine McMeekin ◽  
Djordjo Milovic
Keyword(s):  
Number Field ◽  
Class Number ◽  
Joint Distribution ◽  
Number Fields ◽  
Character Sums ◽  
Prime Ideals ◽  
Odd Degree ◽  
Narrow Class

AbstractLet K be a cyclic number field of odd degree over $${\mathbb {Q}}$$ Q with odd narrow class number, such that 2 is inert in $$K/{\mathbb {Q}}$$ K / Q . We define a family of number fields $$\{K(p)\}_p$$ { K ( p ) } p , depending on K and indexed by the rational primes p that split completely in $$K/{\mathbb {Q}}$$ K / Q , in which p is always ramified of degree 2. Conditional on a standard conjecture on short character sums, the density of such rational primes p that exhibit one of two possible ramified factorizations in $$K(p)/{\mathbb {Q}}$$ K ( p ) / Q is strictly between 0 and 1 and is given explicitly as a formula in terms of the degree of the extension $$K/{\mathbb {Q}}$$ K / Q . Our results are unconditional in the cubic case. Our proof relies on a detailed study of the joint distribution of spins of prime ideals.

scholarly journals A Handy Technique for Fundamental Unit in Specific Type of Real Quadratic Fields

2019 ◽  
Vol 5 (1) ◽  
pp. 495-498
Author(s):  
Özen Özer
Keyword(s):  
Number Theory ◽  
Class Number ◽  
Number Fields ◽  
Fundamental Unit ◽  
Quadratic Fields ◽  
Integral Basis ◽  
Real Quadratic Fields ◽  
Class Number Formula ◽  
Number Formula ◽  
Real Quadratic

AbstractDifferent types of number theories such as elementary number theory, algebraic number theory and computational number theory; algebra; cryptology; security and also other scientific fields like artificial intelligence use applications of quadratic fields. Quadratic fields can be separated into two parts such as imaginary quadratic fields and real quadratic fields. To work or determine the structure of real quadratic fields is more difficult than the imaginary one.The Dirichlet class number formula is defined as a special case of a more general class number formula satisfying any types of number field. It includes regulator, ℒ-function, Dedekind zeta function and discriminant for the field. The Dirichlet’s class number h(d) formula in real quadratic fields claims that we have h\left(d \right).log {\varepsilon _d} = \sqrt {\Delta} {\scr L} \left({1,\;{\chi _d}}\right) for positive d > 0 and the fundamental unit ɛd of {\rm{\mathbb Q}}\left({\sqrt d} \right) . It is seen that discriminant, ℒ-function and fundamental unit ɛd are significant and necessary tools for determining the structure of real quadratic fields.The focus of this paper is to determine structure of some special real quadratic fields for d > 0 and d ≡ 2,3 (mod4). In this paper, we provide a handy technique so as to calculate particular continued fraction expansion of integral basis element wd, fundamental unit ɛd, and so on for such real quadratic number fields. In this paper, we get fascinating results in the development of real quadratic fields.

CLASS NUMBER FORMULAS FOR CERTAIN BICYCLIC BIQUADRATIC NUMBER FIELDS AS FINITE SUMS INVOLVING JACOBI ELLIPTIC FUNCTIONS

2012 ◽  
Vol 08 (05) ◽  
pp. 1257-1270
Author(s):  
M. A. GÓMEZ-MOLLEDA ◽  
JOAN-C. LARIO
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Elliptic Functions ◽  
Number Fields ◽  
Jacobi Elliptic Functions ◽  
Classical Class ◽  
Class Number Formula ◽  
Number Formula ◽  
Finite Sums ◽  
Finite Sum

We give formulas for the class numbers of bicyclic biquadratic number fields containing an imaginary quadratic field of class number one. The class number is expressed as a finite sum in terms of the basic Jacobi elliptic functions, playing a similar role as the trigonometric sine in Dirichlet classical class number formula.

On the Iwasawa λ-invariant of the cyclotomic ℤ2-extension of ℚ(pq) and the 2-part of the class number of ℚ(pq,2 + 2)

2020 ◽  
pp. 1-28
Author(s):  
Naoki Kumakawa
Keyword(s):  
Upper Bound ◽  
Class Number ◽  
Number Fields ◽  
Prime Numbers

In this paper, we study the Iwasawa [Formula: see text]-invariant of the cyclotomic [Formula: see text]-extension of [Formula: see text], where [Formula: see text] are distinct odd prime numbers satisfying certain arithmetic conditions. Moreover, we obtain an upper bound of the [Formula: see text]-part of the class number of certain quartic number fields by calculating the Sinnott index explicitly.

scholarly journals On a class of insoluble binary quadratic diophantine equations

1991 ◽  
Vol 123 ◽  
pp. 141-151 ◽  
Author(s):  
Franz Halter-Koch
Keyword(s):  
Diophantine Equation ◽  
Class Number ◽  
Diophantine Equations ◽  
Number Fields ◽  
Quadratic Number ◽  
Number Problem ◽  
Quadratic Number Fields ◽  
Quadratic Diophantine Equation ◽  
Real Quadratic

The binary quadratic diophantine equationis of interest in the class number problem for real quadratic number fields and was studied in recent years by several authors (see [4], [5], [2] and the literature cited there).

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