Euclidean minima in degree at least four

Determine the Euclidean minima $M(K)$ and $M_2(K)$ for the quartic fields $K=\mathbb{Q}(\sqrt{2+\sqrt2})$ and $K=\mathbb{Q}(\sqrt{3+\sqrt2})$, including whether the conjectured values are correct.

Background

The paper states that, at the time of writing, Euclidean minima had not been determined for number fields of degree at least four. It singles out two quartic fields for which Cohn and Deutsch conjectured explicit first and second minima.

References

For number fields of degree $\ge 4$ no Euclidean minima have hitherto been determined; only Cohn and Deutsch have conjectured that for $K = (\sqrt{2+\sqrt{2})$ we have $M(K) = \frac{1}{2}$ and $M_2(K) = \frac{1}{4}$, while for $K = (\sqrt{3+\sqrt{2})$ they have conjectured that $M(K) = \frac{1}{2}$ and $M_2(K) = \frac{7}{16}$.

— Euclidean Rings  (2608.23216 - Lemmermeyer, 24 Aug 2026) in Remarks on Section 2