We obtain good estimates on the ranks of universal quadratic forms over Shanks’ family of the simplest cubic fields and several other families of totally real number fields. As the main tool, we characterize all the indecomposable integers in these fields and the elements of the codifferent of small trace. We also determine the asymptotics of the number of principal ideals of norm less than the square root of the discriminant.

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

We study totally positive definite quadratic forms over the ring of integers $\mathcal {O}_K$ of a totally real biquadratic field $K=\mathbb {Q}(\sqrt {m}, \sqrt {s})$ . We restrict our attention to classic forms (i.e. those with all non-diagonal coefficients in $2\mathcal {O}_K$ ) and prove that no such forms in three variables are universal (i.e. represent all totally positive elements of $\mathcal {O}_K$ ). Moreover, we show the same result for totally real number fields containing at least one non-square totally positive unit and satisfying some other mild conditions. These results provide 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 $\mathcal {O}_K$ ; we prove several new results about their properties.

In 2016, Jang and Kim stated a conjecture about the norms of indecomposable integers in real quadratic number fields Q √ D where D > 1 is a squarefree integer. Their conjecture was later disproved by Kala for D ≡ 2 mod 4. We investigate such indecomposable integers in greater detail. In particular, we find the minimal D in each congruence class D ≡ 1, 2, 3 mod 4 that provides a counterexample to the Jang-Kim Conjecture; provide infinite families of such counterexamples; and state a refined version of the Jang-Kim conjecture. Lastly, we prove a slightly weaker version of our refined conjecture that is of the correct order of magnitude, showing the Jang-Kim Conjecture is only wrong by at most O √ D .

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