Irrationality of finite logarithms in a congruence-class ad\`ele ring

Finite logarithms of non-zero rational numbers can be defined in the "poor man's ad\`{e}le ring" ${\mathcal A}$ by Fermat quotients modulo sufficiently large primes. This ring contains $\mathbb{Q}$ and outside trivial cases, Matsusaka and Seki have shown that finite logarithms cannot take non-zero rational values in ${\mathcal A}$. Furthermore, a theorem of Silverman shows they are not zero, assuming the $abc$-conjecture. We extend these results to primes restricted to arithmetic progressions of the form $p\equiv 1\bmod m$ by relating Fermat quotients to values of cyclotomic polynomials and their logarithmic derivatives. As an application we show that, subject to the $abc$-conjecture, finite logarithms cannot be quadratic irrational in ${\mathcal A}$ in an appropriate sense.