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Let $f(x)$ be an irreducible polynomial with integer coefficients of degree at least two. Hooley proved that the roots of the congruence equation $f(x)equiv 0mod n$ is uniformly distributed. as a parallel of Hooleys 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.
Let $p>3$ be a prime, and let $(frac{cdot}p)$ be the Legendre symbol. Let $binmathbb Z$ and $varepsilonin{pm 1}$. We mainly prove that $$left|left{N_p(a,b): 1<a<p text{and} left(frac apright)=varepsilonright}right|=frac{3-(frac{-1}p)}2,$$ where $N_p(
We associate to every algebraic number field a hyperbolic surface lamination and an external fundamental group: the latter a generalization of the fundamental germ that necessarily contains external (not first order definable) elements. The external
We give the complete proof of a conjecture of Georges Gras which claims that, for any extension $K/k$ of number fields in which at least one infinite place is totally split, every ideal $I$ of $K$ principalizes in the compositum $Kk^{ab}$ of $K$ with the maximal abelian extension $k^{ab}$ of $k$
Recently, Bruinier and Ono classified cusp forms $f(z) := sum_{n=0}^{infty} a_f(n)q ^n in S_{lambda+1/2}(Gamma_0(N),chi)cap mathbb{Z}[[q]]$ that does not satisfy a certain distribution property for modulo odd primes $p$. In this paper, using Rankin-C
For each odd prime $p$, we conjecture the distribution of the $p$-torsion subgroup of $K_{2n}(mathcal{O}_F)$ as $F$ ranges over real quadratic fields, or over imaginary quadratic fields. We then prove that the average size of the $3$-torsion subgroup