Biquadratic Fields Research Articles

Real quadratic fields with class numbers divisible by five

Conditions are given for a real quadratic field to have class number divisible by five. If 5 does not divide m, then a necessary condition for 5 to divide the class number of the real quadratic field with conductor m or 5m is that 5 divide the class number of a certain cyclic biquadratic field with conductor 5m. Conversely, if 5 divides the class number of the cyclic field, then either one of the quadratic fields has class number divisible by 5 or one of their fundamental units satisfies a certain congruence condition modulo 25.

Dyadotropic Polynomials

Polynomials which tend to represent powers of two arise in connection with certain problems of class field theory of dihedral biquadratic fields. The availability of independent units is an immediate consequence for an infinitude of parametrized cases. An exhaustive search for such types of polynomials is made by use of computer.

Integral bases for bicyclic biquadratic fields over quadratic subfields

A classical question of algebraic number theory is, does an algebraic number field K have an integral basis over a subfield fc? A complete and explicit answer to the above question is given here when K is a bicyclic biquadratic number field and k is a quadratic subfield. Moreover, an explicit integral basis is given for K/k whenever one exists. In the cases where k is imaginary or k is real and has a unit of norm - 1, the conditions involve only rational congruences. When k is real and the fundamental unit of e has norm + 1, the conditions sometimes involve e.

Dyadotropic polynomials

Polynomials which tend to represent powers of two arise in connection with certain problems of class field theory of dihedral biquadratic fields. The availability of independent units is an immediate consequence for an infinitude of parametrized cases. An exhaustive search for such types of polynomials is made by use of computer.

The imaginary bicyclic biquadratic fields with class-number 1.

The imaginary bicyclic biquadratic fields with class-number 1.

Integers of Biquadratic Fields

Let Q denote the field of rational numbers. If m, n are distinct squarefree integers the field formed by adjoining √m and √n to Q is denoted by Q(√m, √n). Since Q(√m, √n) = Q(√m, √n) and √m + √n has for its unique minimal polynomial x4 —2(m + n)x2 + (m - n)2, Q(√m, √n) is a biquadratic field over Q. The elements of Q(√m, √n) are of the form a0 + a1√m + a2√n + a3√mn, where a1, a2, a3 ∊ Q. Any element of Q(√m, √n) which satisfies a monic equation of degree ≥ 1 with rational integral coefficients is called an integer of Q(√m, √n). The set of all these integers is an integral domain. In this paper we determine the explicit form of the integers of Q(√m, √n) (Theorem 1), an integral basis for Q(√m, √n) (Theorem 2), and the discriminant of Q(√m, √n) (Theorem 3).

A gauss bound for a class of biquadratic fields

A bound is obtained for the minimal norm of the ideals in an ideal class of the number field K, where K is a quadratic extension of a Euclidean imaginary quadratic field.

Note on fields of small discriminant

for totally real fields of degree (g - 1)/2, where g is a prime >3, the smallest discriminant is g(g-3)I2, achieved for R (cos 27r/g), where R is the field of rational numbers. (This is the maximal real sub-field in the field of the gth roots of unity.) This conjecture was often made implicitly even before its verification [1] by Davenport for g =7 (the case g =5 having been trivial). We disprove this for those g, such as 13, 37, etc., for which (g+1)/2 is also prime. We use relative quadratic fields built on base fields of small discriminant, a device used previously by Mayer [3 ] in his work on biquadratic fields. The counterexamples are constructed by adjoining to the totally real base-field k the quantity A1/2, where y is a totally positive integer

Diophantine equations reducible in biquadratic fields

Diophantine equations reducible in biquadratic fields