Computing the decomposition of primes p and p-adic absolute values in semi-simple algebras over \({\mathbb{Q}}\) (Q1092109)

Let f be a monic separable polynomial over the rational integers \({\mathbb{Z}}\) and p be a rational prime. We set up an algorithm which yields the extended p-adic values of a given element \(\alpha\) in the semi-simple algebra \(A_ f={\mathbb{Q}}[X]/f\). Firstly, we determine the decomposition of p in \(A_ f\) and an approximate factorization of f over the p-adic completion \({\mathbb{Q}}_ p\) by means of the ORDMAX algorithm of Ford- Zassenhaus. Then it is an easy matter to calculate the values of \(\alpha\). In case it would be necessary to improve the factorization, we thirdly give a modification of the Hensel-Zassenhaus factorization method which works even if the factors are not relatively prime modulo p. Finally, some examples are given at the end of the paper.