AI 中文总结
该数学研究证明了无穷大素数模整数环中，与Tonelli–Shanks算法相关的多个数论对象的像在有理数域上具有超越性。
AI 中文摘要
对于素数p，我们将k_p记为p-1的最大奇因子。Tonelli–Shanks算法是一种经典算法，用于计算模p的二次剩余n∈ℤ的平方根，其形式为n^((k_p+1)/2)（我们称之为n的Tonelli–Shanks幂）与一个二次非剩余模p的幂所构成的修正项的乘积。我们证明了以下对象在无穷大素数模整数环𝒜中的像在ℤ上的超越性：p-1的最大奇因子k_p、任意单射映射f:ℤ→ℤ与整除p-1的最大2次幂(p-1)/k_p的复合f((p-1)/k_p)、任意n∈ℤ∖{-1,0,1}的Tonelli–Shanks幂n^((k_p+1)/2)，以及此类n对应的Tonelli–Shanks算法修正项。
英文摘要
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 $\mathscr{A}$ 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 $f \colon \mathbb{Z} \to \mathbb{Z}$ 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 $n$.
