# How to effectively compute fundamental units in rings?

Consider the ring of integers of $K=\mathbb{Q}(\sqrt 2,\sqrt 3)$. By Dirichlet's unit theorem the units of $\mathcal O_K$ have rank 3, so they are expressible as $\pm u_1^au_2^bu_3^c$ for suitable units $u_1,u_2,u_3$ and $a,b,c\in\mathbb Z$. I found three units which are not expressible as powers of each other: $1+\sqrt 2$, $2+\sqrt 3$ and $\sqrt 3+\sqrt 2$ but how do I guarantee that none of these units is a power of another unit i.e $u^n=1+\sqrt 2$ or the same for the other two? For fields of the form $\mathbb Q(\sqrt m, \sqrt n)$ for coprime squarefree positive integers $m,n$ is it always sufficient to examine the minimal nontrivial solutions to $x^2-my^2=\pm 1$, $x^2-ny^2=\pm 1$ and $x^2-mny^2=\pm 1$?

• I know the papers of M. Pohst for the effective computation of fundamental units in general, e.g., see here. For real biquadratic number fields this should simplify; perhaps this is already done somewhere. I found more explicit examples here. Have you see the thesis of N. Jeans? Nov 17, 2017 at 19:49
• You can search for $3$ units $u_1,u_2,u_2$ with $U_j = \log(\sigma_1(u_j),\ldots,\log\sigma_4(u_j))$ linearly independent and check if $\det(U)$ matches the regulator obtained from $\zeta_K$. Nov 17, 2017 at 21:13