Universal quadratic forms and indecomposables over biquadratic fields

Abstrakt

The aim of this article is to study (additively) indecomposable algebraic integers $\mathcal O_K$ of biquadratic number fields $K$ and universal totally positive quadratic forms with coefficients in $\mathcal O_K$. There are given sufficient conditions for an indecomposable element of a quadratic subfield to remain indecomposable in the biquadratic number field $K$.

Furthermore, estimates are proven which enable algorithmization of the method of escalation over $K$. These are used to prove, over two particular biquadratic number fields $\BQ 23$ and $\BQ{6}{19}$, a lower bound on the number of variables of a universal quadratic forms.

Klíčová slova

• indecomposable integer
• universal quadratic form
• biquadratic number field