Explicit form of Cassels' $p$-adic embedding theorem for number fields

TL;DR

This paper provides an explicit form of Cassels' $p$-adic embedding theorem for number fields and refines it for cyclotomic fields, also offering bounds on primes related to polynomial roots modulo p.

Contribution

It presents a general explicit form of Cassels' $p$-adic embedding theorem and refines it for cyclotomic fields, including bounds on primes for polynomial roots modulo p.

Findings

Explicit form of Cassels' $p$-adic embedding theorem for number fields

Refined form for cyclotomic fields

Unconditional upper bounds for primes related to polynomial roots

Abstract

In this paper, we mainly give a general explicit form of Cassels' $p$-adic embedding theorem for number fields. We also give its refined form in the case of cyclotomic fields. As a byproduct, given an irreducible polynomial $f$ over $Z$, we give a general unconditional upper bound for the smallest prime number $p$ such that $f$ has a simple root modulo $p$.