We study Euclidean ideal classes in real biquadratic fields and obtain unconditional existence results via genus theory. Lenstra showed (assuming the Generalized Riemann Hypothesis) that a number field with unit rank at least one admits a Euclidean ideal precisely when its class group is cyclic; subsequent work has aimed to remove the GRH hypothesis in special families. Focusing on real biquadratic fields $K=\mathbb{ Q}\left(\sqrt{ d_1},\sqrt{ d_2}\right)$ with $2\nmid d_1d_2$, we prove that if the class group $\mathrm{ Cl}_K$ is cyclic and the Hilbert class field $H(K)$ is abelian over $\mathbb{ Q}$, then $K$ contains a Euclidean ideal class (unconditionally). We also analyse the distribution of genus numbers in a natural family of biquadratic fields and, using these statistics, show that the set of biquadratic fields admitting a Euclidean ideal has density zero.
Categories: math.NT