Universal quadratic forms and indecomposables over biquadratic fields

Abstract

The aim of this article is to study (additively) indecomposable algebraic integers  of biquadratic number fields and universal totally positive quadratic forms with coefficients in  . There are given sufficient conditions for an indecomposable element of a quadratic subfield to remain indecomposable in the biquadratic number field . Furthermore, estimates are proven which enable algorithmization of the method of escalation over . These are used to prove, over two particular biquadratic number fields ℚ ( √ 2,, a lower bound on the number of variables of a universal quadratic forms. K E Y W O R D S biquadratic number field, indecomposable integer, universal quadratic form M S C ( 2 0 1 0 ) 11E12, 11E25, 11R04 540

There are no universal ternary quadratic forms over biquadratic fields

“…Some general statements can be found in the work of Brunotte [4]. Considering biquadratic fields, we can draw on the results of [6], which we extend in this paper. In particular, we prove the following theorem.…”

“…The following lemma generalizes [6,Lemma 4.1]. Note that in the statement, the meaning of n 3 is given by Convention 2.2 as…”

“…In particular, the statement of part (1) is covered by Propositions 4.9 and 4.10, and part (2) by Proposition 4.11. Part (3) follows from Example 4.7; note that this example provides an indecomposable algebraic integer from a quadratic field, which decomposes in a biquadratic field, and thus solving an open question from [6].…”

There are no universal ternary quadratic forms over biquadratic fields

“…Bearing this notation in mind, we can give the following necessary condition for the totally positive elements of O K . This lemma was derived in [C+,Lemma 3.1].…”

“…In biquadratic fields, we do not have such a characterization of indecomposable integers. The only to us known result is [C+,Th. 2.1] which claims that under certain conditions, the indecomposables from quadratic subfields remain indecomposable in the biquadratic field.…”