Riemann hypothesis and finding roots over finite fields

• Authors: M-D A Huang
• Venue: STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
• Year: 1985

It is shown that assuming Generalized Riemann Hypothesis, the roots of ƒ(x) = O mod p, where p is a prime and f(x) is an integral Abilene polynomial can be found in deterministic polynomial time. The method developed for solving this problem is also applied to prime decomposition in Abelian number fields, and the following result is obtained: assuming Generalized Riemann Hypotheses, for Abelian number fields K of finite extension degree over the rational number field Q, the decomposition pattern of a prime p in K, i.e. the ramification index and the residue class degree, can be computed in deterministic polynomial time, providing p does not divide the extension degree of K over Q. It is also shown, as a theorem fundamental to our algorithm, that for q, p prime and m the order of p mod q, there is a q-th nonresidue in the finite field Fpm that can be written as ao + a1w + … + am-1wm-1, where |a1| ≤ cq2 log2(pq), c is an absolute effectively computable constant, and 1, w, …, wm-1 form a basis of Fpm over Fp. More explicitly, w is a root of the q-th cyclotomic polynomial over Fp. This result partially generalizes, to finite field extensions over Fp, a classical result in number theory stating that assuming Generalized Riemann Hypothesis, the least q-th nonresidue mod p for p,q prime and q dividing p - t is bounded by c log2p, where c is an absolute, effectively computable constant.