Factorization of natural numbers in some quadratic number fields

Computation of class numbers of quadratic number fields

We explain how one can dispense with the numerical computation of approximations to the transcendental integral functions involved when computing class numbers of quadratic number fields. We therefore end up with a simpler and faster method for computing class numbers of quadratic number fields. We also explain how to end up with a simpler and faster method for computing relative class numbers ...

Class Numbers of Real Quadratic Number Fields by Ezra Brown

This article is a study of congruence conditions, modulo powers of two, on class number of real quadratic number fields Q(vu), for which d has at most thtee distinct prime divisors. Techniques used are those associated with Gaussian composition of binary quadratic forms. 1. Let hid) denote the class number of the quadratic field Qi\ß) and let h id) denote the number of classes of primitive bina...

Uniqueness of Factorization in Quadratic Fields

Real Quadratic Number Fields

a4 + 1 a5 + .. . will see that a less wasteful notation, say [ a0 , a1 , a2 , . . . ] , is needed to represent it. Anyone attempting to compute the truncations [ a0 , a1 , . . . , ah ] = ph/qh will be delighted to notice that the definition [ a0 , a1 , . . . , ah ] = a0 + 1/[ a1 , . . . , ah ] immediately implies by induction on h that there is a correspondence ( a0 1 1 0 ) ( a1 1 1 0 ) · · · (...

Quadratic Fields and Transcendental Numbers

Mohammad Zaki, MN State Univ, Mankato We define an algebraic number α as a root of an algebraic equation, a0∗x+a1∗x+· · ·+an−1∗x+an = 0 where ao, a1, · · · , an are rational integers, not all zero. We say α is an algebraic integer if a0 = 0. If an algebraic number α satisfies an algebraic equation of degree n with rational coefficients, and none of lower degree, then we say α is of degree n. If...