There are no universal ternary quadratic forms over biquadratic fields

Abstrakt

We study totally positive definite quadratic forms over the ring of integers O_K of a totally real biquadratic field K= Q(sqrt(m), sqrt(s)). We restrict our attention to classical forms (i.e., those with all non-diagonal coefficients in 2O_K) and prove that no such forms in three variables are universal (i.e., represent all totally positive elements of O_K).

This provides further evidence towards Kitaoka's conjecture that there are only finitely many number fields over which such forms exist. One of our main tools are additively indecomposable elements of O_K; we prove several new results about their properties.

Klíčová slova

• ndecomposable integer
• universal quadratic form
• ternary quadratic form
• biquadratic number field