On computing subfields. A detailed description of the algorithm

An efficient algorithm for the computation of Galois automorphisms

“…In practice, it may also be convenient to choose Σ so that R is squarefree modulo a specific prime or prime power. Note that Lemma 43 in [7] is false, since Lemma 42 applies only to irreducible polynomials.…”

“…A deterministic way to find Σ is given in [7,Lemma 32]. To reduce the needed accuracy on the roots, it is better to find a polynomial Σ of small degree.…”

Topics in computational algebraic number theory

“…. , w n ) 23 of the maximal order O K . Then we may identify K with Q n : an element n i=1 x i w i in K is represented as the column vector x := (x 1 , .…”

“…This P is still large: N K/Q (disc(P )) has 2628 decimal digits. We compute the absolute extension (< 10ms), use a polynomial reduction algorithm (1min 45s) and search for subfields [23] (< 10ms) to eventually produce the polynomial…”

Explicit Galois Realization of Transitive Groups of Degree up to 15

“…When a number field K has non-trivial subfields, very often the following procedure is useful: We remark that subfield computation can be done with algorithms described in Klüners and Pohst (1997) and Klüners (1998).…”