On Reconstruction of Algebraic Numbers

Factoring Polynomials Over Local Fields

“…If the residue class field k is finite we can use the algorithms of Berlekamp (1970), Cantor and Zassenhaus (1981), or one of the many improvements of these algorithms, Kaltofen and Shoup (1998) for example. If k is the completion of a function field over a number field then polynomials over k can be factored using the algorithms for factoring polynomials over number fields by Trager (1976), Pohst (1999), Roblot (2000), or Fieker and Friedrichs (2000).…”

A relative van Hoeij algorithm over number fields

“…Choosing T 2 yields smaller bounds (only slightly so, provided (ω i ) is LLL-reduced), but would force us to compute embeddings to a huge accuracy to avoid stability problems during the LLL reduction. This idea is due to Fieker and Friedrichs [12]. Remark 3.14.…”

Topics in computational algebraic number theory

“…In many applications, it is advantageous to use non-Archimedean embeddings K → K ⊗ Q Q p = ⊕ p|p K p which is isomorphic to Q n p as a Q p -vector space. This cancels rounding errors, as well as stability problems in the absence of divisions by p. In some applications (e.g., automorphisms [1], factorization of polynomials [3,18]), a single embedding K → K p is enough, provided an upper bound for α is available.…”