Homomorphism Counts Quadratic Residues

Abstract

We prove that the ratio $\varrho(n)=\phi(n)/2^{\omega(n)}$ 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 $(\mathbb Z/n \mathbb Z)^*$, 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 $n\equiv2\pmod{4}$ and $n\notin \mathscr{E}$. Divisibility of this ratio fails precisely when the odd part of~$n$ is composed entirely of Gaussian primes, and the exception set satisfies $|\mathscr{E}\cap[2,x]|\sim Cx/\sqrt{\log x}$ with explicit constant $C\approx0.279$. 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.