Homomorphism Counts Quadratic Residues

Abstract

We prove that the ratio $\varrho (n)=\varphi (n)/2^{\omega (n)}$ of surjective group to ring homomorphism counts between finite cyclic rings admits three simultaneous interpretations that have not previously been connected. It equals the order of the group of squares in $(\mathbb{Z}/n\mathbb{Z}{)}^{\ast }$, the degree $[\mathbb{Q}({\zeta }_n):K_n]$ of the $n$-th cyclotomic field over its maximal biquadratic subfield, and a product determined by the nonzero quadratic residue counts in the odd prime-power components of $n$, equal to that product when $n$ is odd or $4\mid n$, and half that product when $n\equiv 2(\mathrm{mod}4)$ and $n\notin \mathcal{E}$. Divisibility of this ratio fails precisely when the odd part of~$n$ is composed entirely of Gaussian primes, and the exception set satisfies $|\mathcal{E}\cap [2,x]|\sim Cx/\sqrt{\log x}$ with explicit constant $C\approx 0.279$. The cyclotomic interpretation is new and yields, in particular, an algebraic proof that $[\mathbb{Q}({\zeta }_n):K_n]$ is always an integer: this is the tower law applied to a field degree, recovering the divisibility result without any case analysis.