Quadratic forms over formally real fields with eight square classes

Complete classification of formally real fields with 8 square classes with respect to the behaviour of quadratic forms is given. Two fields F and K are equivalent with respect to quadratic forms if the quadratic form schemes of the two fields are isomorphic or in other words, if the Witt rings W(F) and W(K) are isomorphic. It is shown here that for formally real fields with 8 square classes there are exactly seven possible quadratic form schemes and for each of the seven schemes a formally real field with 8 square classes and the given scheme is constructed.     

Invariants for quadratic forms

Simple systems of invariants for rational and integral quadratic forms are given, and those for rational forms are proved complete in an elementary way. Some noninvariants of quadratic forms appear, but are not concerned with invariants of objects other than quadratic forms. Our treatment of noninvariants of objects other than quadratic forms is minimal, and it is here that there is most room for further investigation.     

Universal octonary diagonal forms over some real quadratic fields

In this paper, we will prove there are infinitely many integers n such that n ²— 1 is square-free and admits universal octonary diagonal quadratic forms. Received: November 2, 1998.     

Extending orderings on formally real fields

Archiv der Mathematik -     

Theta functions of indefinite quadratic forms over real number fields

We define theta functions attached to indefinite quadratic forms over real number fields and prove that these theta functions are Hilbert modular forms by regarding them as specializations of symplectic theta functions. The eighth root of unity which arises under modular transformations is determined explicitly.

     

Overpartitions and real quadratic fields

It is shown that counting certain differences of overpartition functions is equivalent to counting elements of a given norm in appropriate real quadratic fields.     

Normal forms for real quadratic forms

We investigate the non-diagonal normal forms of a quadratic form on , in particular for n = 3. For this case it is shown that the set of normal forms is the closure of a 5-dimensional submanifold in the 6-dimensional Grassmannian of 2-dimensional subspaces of . Received: 27 June 2008     

The fan-pall imbedding theorem over formally real fields

The converse of the Cauchy interlacing theorem, relating eigenvalues of a symmetric real matrix and eigenvalues of a principal submatrix, first proved by Fan and Pall, is extended to the case of symmetric matrices with entries in an arbitrary formally real field.     

On some Hasse Principles over formally real fields

Mathematische Zeitschrift -     

The fan-pall imbedding theorem over formally real fields

The converse of the Cauchy interlacing theorem, relating eigenvalues of a symmetric real matrix and eigenvalues of a principal submatrix, first proved by Fan and Pall, is extended to the case of symmetric matrices with entries in an arbitrary formally real field.     

On real quadratic function fields

Let q be a power of an odd prime number p and K:=Fq(T) be the rational function field with a fixed indeterminate T. For P a prime of K, let be the maximal real subfield of the Pth-cyclotomic function field and its ring of integers. We prove that there exists infinitely many primes P of even degree such that the cardinal of the ideal class group is divisible by q. We prove also an analogous result for imaginary extensions.     

Finiteness of real quadratic fields which admit positive integral diagonal septanary universal forms

In this paper, we will prove if D is large enough, there are no positive integral diagonal septanary universal quadratic forms over . Received: 13 November 1997 / Revised version: 17 November 1998     

Representation of integers by positive-definite quadratic forms in totally real algebraic number fields

The results of Gundlach (K. B. Gundlach, Acta Math.,92, 309–345 (1954)) are extended to forms with an odd number of variables. We obtain for alln4 asymptotic formulae for the number of integral representations of numbers by totally positiven-ary forms in totally real algebraic number fields. The remainder term is of the form . The necessary concepts and results about Hilbert modular forms of half integral weight are developed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 32–46, 1983.     

On units of real quadratic fields

Let ∈ = (t + u(d)2/2 be a unit of Q((d)1/2), whose norm = 1. We investigate the properties of the factors of the number u_n which is defined by ∈ⁿ = (t_n + u_n(d)1/2)/2.     

Caliber number of real quadratic fields

We obtain lower bound of caliber number of real quadratic field using splitting primes in K. We find all real quadratic fields of caliber number 1 and find all real quadratic fields of caliber number 2 if d is not 5 modulo 8. In both cases, we don't rely on the assumption on ζK(1/2).     

Fibonacci numbers and real quadratic p-rational fields

Periodica Mathematica Hungarica - We characterize p-rational real quadratic fields in terms of generalized Fibonacci numbers. We then use this characterization to give numerical evidence to a...