Dihedral fields with Bloch lattice index five and quadratic subfield Q(i)

Abstract

We construct infinitely many Galois extensions $M/\mathbb{Q}$ with dihedral group of order ten, quadratic subfield $\mathbb{Q}(i)$, and Bloch lattice index $Q_2(M)=5$. More precisely, we specify a Chebotarev set of rational primes of density $1/50$. For every prime $\ell$ in this set, at least two of four mixed cyclic quintic extensions of $\mathbb{Q}(i)$ of conductor $(25\ell)$ have this index and satisfy $K_2(O_M)(5)\simeq(\mathbb{Z}/5)^2$. A two-dimensional cup-product pencil selects the extensions. Generalized Bockstein maps, integral co-descent, and a motivic genus formula then determine the tame kernel and the subfield Bloch-lattice index. The result is unconditional.