Transcendence of Tonelli--Shanks Power Modulo Infinitely Large Primes

For a prime number $p$, we denote by $k_p$ the greatest odd divisor of $p-1$. Tonelli–Shanks algorithm is a classical algorithm to compute a square root of a quadratic residue $n\in \mathbb{Z}$ modulo $p$ as the product of $n^{\frac{k_p+1}{2}}$, which we call {\it Tonelli–Shanks power of $n$}, and a correction term given as a power of a quadratic non-residue modulo $p$. We prove the transcendence over $\mathbb{Q}$ of the images of the following in the ring $\mathcal{A}$ of integers modulo infinitely large primes: the greatest odd divisor $k_p$ of $p-1$, the composite $f(\frac{p-1}{k_p})$ of any injective map $f\text{\hspace{0.17em}⁣}:\mathbb{Z}\to \mathbb{Z}$ and the greatest $2$-power $\frac{p-1}{k_p}$ dividing $p-1$, Tonelli–Shanks power $n^{\frac{{}_p+1}{2}}$ of any $n\in \mathbb{Z}\setminus \{-1,0,1\}$, and the correction term of Tonelli–Shanks algorithm for such $n$.