Distribution of residues of an algebraic number modulo ideals of degree one

TL;DR

This paper extends Hooley's uniform distribution theorem from integers to algebraic numbers modulo degree one ideals, demonstrating the uniformity of roots of polynomial congruences and discussing n-adic digit distributions.

Contribution

It generalizes the uniform distribution of polynomial roots from integers to algebraic numbers modulo degree one ideals, and explores implications for polynomial systems and n-adic expansions.

Findings

Roots of polynomial congruences are uniformly distributed modulo degree one ideals.

Distribution of digits in n-adic expansions of algebraic numbers is analyzed.

Theoretical framework connecting algebraic number residues and ideal theory.

Abstract

Let $f(x)$ be an irreducible polynomial with integer coefficients of degree at least two. Hooley proved that the roots of the congruence equation $f(x)\equiv 0\mod n$ is uniformly distributed. as a parallel of Hooley's theorem under ideal theoretical setting, we prove the uniformity of the distribution of residues of an algebraic number modulo degree one ideals. Then using this result we show that the roots of a system of polynomial congruences are uniformly distributed. Finally, the distribution of digits of n-adic expansions of an algebraic number is discussed.