7,603 research outputs found
Computation of Galois groups of rational polynomials
Computational Galois theory, in particular the problem of computing the
Galois group of a given polynomial is a very old problem. Currently, the best
algorithmic solution is Stauduhar's method. Computationally, one of the key
challenges in the application of Stauduhar's method is to find, for a given
pair of groups H<G a G-relative H-invariant, that is a multivariate polynomial
F that is H-invariant, but not G-invariant. While generic, theoretical methods
are known to find such F, in general they yield impractical answers. We give a
general method for computing invariants of large degree which improves on
previous known methods, as well as various special invariants that are derived
from the structure of the groups. We then apply our new invariants to the task
of computing the Galois groups of polynomials over the rational numbers,
resulting in the first practical degree independent algorithm.Comment: Improved version and new titl
On intersective polynomials with non-solvable Galois group
We present new theoretical results on the existence of intersective
polynomials with certain prescribed Galois groups, namely the projective and
affine linear groups and as well as the affine
symplectic groups . For
further families of affine groups, existence results are proven conditional on
the existence on certain tamely ramified Galois extensions. We also compute
explicit families of intersective polynomials for certain non-solvable groups.Comment: 16 page
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