5-Class towers of cyclic quartic fields arising from quintic reflection

Abstract

Let $$\zeta _5$$ be a primitive fifth root of unity and $$d\ne 1$$ be a quadratic fundamental discriminant not divisible by $$5$$ . For the $$5$$ -dual cyclic quartic field $${M}={\mathbb {Q}}((\zeta _5-\zeta _5^{-1})\sqrt{d})$$ of the quadratic fields $${k}_1={\mathbb {Q}}(\sqrt{d})$$ and $${k}_2={\mathbb {Q}}(\sqrt{5d})$$ in the sense of the quintic reflection theorem, the possibilities for the isomophism type of the Galois group $$\mathrm {G}_5^{(2)}{{M}}=\mathrm {Gal}({M}_5^{(2)}/{M})$$ of the second Hilbert $$5$$ -class field $${M}_5^{(2)}$$ of $${M}$$ are investigated, when the $$5$$ -class group $$\mathrm {Cl}_5(M)$$ is elementary bicyclic of rank two. Usually, the maximal unramified pro- $$5$$ -extension $${M}_5^{(\infty )}$$ of $$M$$ coincides with $${M}_5^{(2)}$$ already. The precise length $$\ell _5{M}$$ of the $$5$$ -class tower of $${M}$$ is determined, when $$\mathrm {G}_5^{(2)}{{M}}$$ is of order less than or equal to $$5^5$$ . Theoretical results are underpinned by the actual computation of all $$83$$ , respectively $$93$$ , cases in the range $$0<d<10^4$$ , respectively $$-2\cdot 10^5<d<0$$ .