A Note on the Class-Numbers of Algebraic Number Fields

Ankeny, N. C.; Brauer, Richard; Chowla, S.

doi:10.2307/2372483

Cited by 23 publications

(21 citation statements)

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“…For a number field K of signature (r 1 , r 2 ) and degree n = r 1 + 2r 2 , let τ (K) = r 1 /n ∈ I be the proportion of its embeddings which are real. Let us call τ (K) the infinity type of K. Number fields of degree n ≥ 1 and infinity type t ∈ I exist if and only if nt and n(1 − t)/2 are integral (see, for example, Ankeny et al, 1956). For such n and t, let R t (n) be the minimal root discriminant for number fields of degree n and infinity type t. (The root discriminant rd K of K is defined by rd K = |d K | 1/n where d K is the discriminant of K.) Define a function α on I by…”

Section: Introductionmentioning

confidence: 99%

Tamely Ramified Towers and Discriminant Bounds for Number Fields—II

Hajir

Maire

2002

Journal of Symbolic Computation

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The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R 0 (2m) be the minimal root discriminant for totally complex number fields of degree 2m, and put α 0 = lim infm R 0 (2m). Define R 1 (m) to be the minimal root discriminant of totally real number fields of degree m and put α 1 = lim infm R 1 (m). Assuming the Generalized Riemann Hypothesis, α 0 ≥ 8πe γ ≈ 44.7, and, α 1 ≥ 8πe γ+π/2 ≈ 215.3. By constructing number fields of degree 12 with suitable properties, we give the best known upper estimates for α 0 and α 1 : α 0 < 82.2, α 1 < 954.3.

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Section: Introductionmentioning

confidence: 99%

Tamely Ramified Towers and Discriminant Bounds for Number Fields—II

Hajir

Maire

2002

Journal of Symbolic Computation

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“…It is easy to prove that under the Artin conjecture and Generalized Riemann Hypothesis (GRH) for L(s, ρ), L (1, For real quadratic fields, this is a classical result of Montgomery and Weinberger [14]. Ankeny, Brauer, and Chowla [1] constructed unconditionally, for any n, r 1 , r 2 , number fields with arbitrarily large discriminants and h K |d K | 1/2− . Under the GRH and Artin conjecture for L(s, ρ), Duke [6] constructed totally real fields of degree n whose Galois closures have the Galois group S n with the largest possible class numbers.…”

Section: Introductionmentioning

confidence: 94%

Application of the Strong Artin Conjecture to the Class Number Problem

Cho¹,

Kim²

2013

Can. j. math.

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Abstract. We construct unconditionally several families of number fields with the largest possible class numbers. They are number fields of degree 4 and 5 whose Galois closures have the Galois group A 4 , S 4 and S 5 . We first construct families of number fields with smallest regulators, and by using the strong Artin conjecture and applying zero density result of Kowalski-Michel, we choose subfamilies of L-functions which are zero free close to 1. For these subfamilies, the L-functions have the extremal value at s = 1, and by the class number formula, we obtain the extreme class numbers.

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