Divisibility Of Class Numbers Research Articles

On divisibility of class number of real Abelian fields of prime conductor

On divisibility of class number of real Abelian fields of prime conductor

Class Number Divisibility in Real Quadratic Function Fields

AbstractLet q be a positive power of an odd prime p, and let Fq(t) be the function field with coefficients in the finite field of q elements. Let denote the ideal class number of the real quadratic function field obtained by adjoining the square root of an even-degree monic . The following theorem is proved: Let n ≧ 1 be an integer not divisible by p. Then there exist infinitely many monic, squarefree polynomials, such that n divides the class number, . The proof constructs an element of order n in the ideal class group.

Asymptotic Results for Class Number Divisibility in Cyclotomic Fields

AbstractLet n ≥3 and m≥3 be integers. Let Kn be the cyclotomic field obtained by adjoining a primitive nth root of unity to the field of rational numbers. Let denote the maximal real subfield of Kn. Let hn (resp., ) denote the class number of Kn (resp., ). For fixed m we show that m divides hn and hn for asymptotically almost all n. Also for those Kn and with a given number of ramified primes, we obtain lower bounds for certain types of densities for m dividing hn and .

Class numbers and a generalized Fermat theorem

Conditions for divisibility of class numbers of algebraic number fields by prime powers are explored and linked to the existence of integer solutions of certain cyclotomic polynomials modulo a given rational integer. Several applications are provided, including a generalization of the Fermat “two-square theorem”.

Divisibility of class numbers of real cyclic extensions of degree 4 over Q.

Article Divisibility of class numbers of real cyclic extensions of degree 4 over Q. was published on January 1, 1979 in the journal Journal für die reine und angewandte Mathematik (volume 1979, issue 307-308).

Generalization of Kummer's criterion for divisibility of class numbers

We give a necessary and sufficient condition for the relative class number of an imaginary field contained in Q(e 2πi/p ϱ ) to be divisible by p. We also give a sufficient condition for the class number of a real field contained in Q(e 2πi/p ϱ ) not to be divisible by p.

A rational genus, class number divisibility, and unit theory for pure cubic fields

The classical genus theory of Gauss has been extended by Hilbert from the quadratic field over the rational field to the Kummer field over the cyclotomic field as a precise analog. Still, many problems involving the computation of pure cubic unit and class number are related to the rational and not the cubic cyclotomic field. It is, therefore, desirable to have (at least partially) a genus theory of cubic characters and an analog of such things as Hilbert's Theorem 90, ambiguous (principal) classes, and class group invariants divisible by 3 and 9,-all based on the rational field. This study shows a strong similarity between pure cubic and quadratic fields.