Following the analogies between knots and primes, we establish relative genus theory for a branched cover of rational homology 3-spheres. Then we formulate analogues of Iwasawa’s theorems on $\mu$-invariants for branched $\mathbf{Z}_{p}$-covers of rational homology 3-spheres, by using relative genus theory.

Let $\zeta_{n}$ be a primitive $n$th root of unity. As is well known, $\mathbf{Z}[\zeta_{n}+\zeta_{n}^{-1}]$ is the ring of integers of $\mathbf{Q}(\zeta_{n}+\zeta_{n}^{-1})$. We give an alternative proof of this fact by using the resultants of modified cyclotomic polynomials.