We show that for all real biquadratic fields not containing $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, $\sqrt{6}$, $\sqrt{7}$ and $\sqrt{13}$, the Pythagoras number of the ring of algebraic integers is at least 6. We also provide an upper bound on the norm and the minimal (codifferent) trace of additively indecomposable integers in some families of these fields.

We show that for $100\%$ of the odd, square free integers $n> 0$ , the $4$ -rank of $\text {Cl}(\mathbb{Q} (i, \sqrt {n}))$ is equal to $\omega _3(n) - 1$ , where $\omega _3$ is the number of prime divisors of n that are $3$ modulo $4$ .