A Fano problem consists of enumerating linear spaces a fixed dimension on variety, generalizing the classical 27 lines cubic surface. Those problems with finitely many have an associated Galois group that acts these and controls complexity computing them in suitable coordinates. These groups were first defined studied by Jordan, who particular considered P3 Recently, Hashimoto Kadets determined...

In the study of Galois theory, after computing a few Galois groups of a given field, it is very natural to ask the question of whether or not every finite group can appear as a Galois group for that particular field. This question was first studied in depth by David Hilbert, and since then it has become known as the Inverse Galois Problem. It is usually posed as which groups appear as Galois ex...

This paper is a chronological survey, with no proofs, of a direction in categorical algebra, which is based on categorical Galois theory and involves generalized central extensions, commutators, and internal groupoids in Barr exact Mal’tsev and more general categories. Galois theory proposes a notion of central extension, and motivates the study of internal groupoids, which is then used as an a...

Let F be a field with charF 6= 2. We show that there are two groups of order 32, respectively 64, such that a field F with char F 6= 2 is nonrigid if and only if at least one of the two groups is realizable as a Galois group over F . The realizability of those groups turns out to be equivalent to the realizability of certain quotients (of order 16, respectively 32). Using known results on conne...

A criterion is given for the solvability of a central Galois embedding problem to go from a projective linear group covering to a vectorial linear

Let f ∈ Q[x] be an irreducible polynomial of degree n. Then the splitting field L ≥ Q of f is a normal extension. We want to determine the Galois group G = Gal(f) = Gal(L/Q) of f which is the group of all field automorphisms of this extension. This task is basic in computational number theory [Coh93] as the Galois group determines a lot of properties of the field extension defined by f . Becaus...

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A technique is described for the nontentative computer determination of the Galois groups of irreducible polynomials with integer coefficients. The technique for a given polynomial involves finding high-precision approximations to the roots of the polynomial, and fixing an ordering for these roots. The roots are then used to create resolvent polynomials of relatively small degree, the linear fa...