Binary quadratic forms represented by a sum of nonzero squares

Quadratic Polynomials Represented by Norm Forms

— Let P (t) ∈ Q[t] be an irreducible quadratic polynomial and suppose that K is a quartic extension of Q containing the roots of P (t). Let NK/Q(x) be a full norm form for the extension K/Q. We show that the variety P (t) = NK/Q(x) 6= 0 satisfies the Hasse principle and weak approximation. The proof uses analytic methods.

Integers Represented by Idoneal Quadratic Forms

There are 65 known positive integers n for which a prime number p prime to D is of the form x2 + ny2 iff p lies in certain congruence classes modulo 4n: these are the idoneal numbers. (It is known that there is at most one further such number and if such a 66th idoneal number exists, then the Generalized Riemann Hypothesis is false.) For n = 1 this is Fermat’s Two Squares Theorem: an odd prime ...

Composition of Binary Quadratic Forms

Applications of quadratic D-forms to generalized quadratic forms

In this paper, we study generalized quadratic forms over a division algebra with involution of the first kind in characteristic two. For this, we associate to every generalized quadratic from a quadratic form on its underlying vector space. It is shown that this form determines the isotropy behavior and the isometry class of generalized quadratic forms.

Division and Binary Quadratic Forms

has only three elements, written h(−23) = 3. There is an binary operation called composition that takes two primitive forms of the same discriminant to a third. Composition is commutative and associative, and makes the set of forms into a group, with identity 〈1, 0,−∆/4〉 for even discriminant and 〈1, 1, (1−∆)/4〉 for odd. From page 49 of Buell [1]: if a form 〈α, β, γ〉 represents a number r primi...