This article provides necessary and sufficient conditions for each group of order 32 to be realizable as a Galois group over an arbitrary field. These conditions, given in terms of the number of square classes of the field and the triviality of specific elements in related Brauer groups, are used to derive a variety of automatic realizability results.
Irreducibility over of a special symmetric form in a variables is proved for .
A necessary and sufficient condition is given for reducibility of a symmetric polynomial whose number of variables is large in comparison to degree.
We show explicit forms of the Bertini-Noether reduction theorem and of the Hilbert irreducibility theorem. Our approach recasts in a polynomial context the geometric Grothendieck good reduction criterion and the congruence approach to HIT for covers of the line. A notion of “bad primes” of a polynomial P ∈ ℚ[T,Y] irreducible over ℚ̅ is introduced, which plays a central and unifying role. For such a polynomial P, we deduce a new bound for the least integer t₀ ≥ 0 such that P(t₀,Y) is irreducible...
Let (F,D) be a differential field with the subfield of constants C (c ∈ C iff Dc=0). We consider linear differential equations (1) , where , and the solution y is in F or in some extension E of F (E ⊇ F). There always exists a (minimal, unique) extension E of F, where Ly=0 has a full system of linearly independent (over C) solutions; it is called the Picard-Vessiot extension of F E = PV(F,Ly=0). The Galois group G(E|F) of an extension field E ⊇ F consists of all differential automorphisms of...