Polynomials with Dp as Galois group

A useful criterion characterizing a monic irreducible polynomial over $\text{Q}$ with Galois group D_p (the dihedral group of order 2p, p: prime) is given by making use of the geometry of D_p, i.e., D_p is the symmetry group of the regular p-gon. We derive explicit numerical examples of polynomials with dihedral Galois groups D₅ and D₇.     

Polynomials with Cyclic Galois Group

It is proved that certain polynomials of degree ?3 have a cyclic group of order n as Galois group over all rational function fields k(t) with characteristic not dividing n. Moreover, the extension fields of k(t) generated by the polynomials have k as precise field of constants, and possess an unramified rational point. For all 3≤?20 with the exceptions of 17 and 19 the polynomials are calculated explicitly     

The Valentiner group as Galois group

We obtain the complete set of solutions to the Galois embedding problem given by the Valentiner group as a triple cover of the alternating group .

     

Bivariate Factorizations via Galois Theory, with Application to Exceptional Polynomials

We present a method for factoring polynomials of the shapef(X) − f(Y), wherefis a univariate polynomial over a fieldk. We then apply this method in the case whenfis a member of the infinite family of exceptional polynomials we discovered jointly with H. Lenstra in 1995; factoringf(X) − f(Y) in this case was posed as a problem by S. Cohen shortly after the discovery of these polynomials.     

On the mutual embeddability of (2k, k, k − 1) and (2k − 1, k, k) quasi-residual designs

The complements of blocks containing a given point in a (2k ? 1, k, k) design, enlarged by this point, and the blocks not containing it, form a (2k ? 1, k, k) design. Likewise, the complements of blocks containing a given point in a (2k, k, k ? 1) design and the blocks not containing it, form a (2k ? 1, k, k) design. In this paper we show that if a quasi-residual (2k ? 1, k, k) design is obtained from an embeddable (2k ? 1, k, k) or (2k, k, k ? 1) design, then it is also embeddable, and describe an example of non-embeddable (12, 6, 5) design such that all (11, 6, 6) designs obtained from it are embeddable.     

Infinitesimal group schemes as iterative differential Galois groups

This article is concerned with Galois theory for iterative differential fields (ID-fields) in positive characteristic. More precisely, we consider purely inseparable Picard-Vessiot extensions, because these are the ones having an infinitesimal group scheme as iterative differential Galois group. In this article we prove a necessary and sufficient condition to decide whether an infinitesimal group scheme occurs as Galois group scheme of a Picard-Vessiot extension over a given ID-field or not. In particular, this solves the inverse ID-Galois problem for infinitesimal group schemes. Furthermore, this gives a tool to tell whether all purely inseparable ID-extensions are in fact Picard-Vessiot extensions.     

Uniqueness of norm on L(G) and C(G) when G is a compact group

Let G be a compact group. If the trivial representation of G is not weakly contained in the left regular representation of G on L₀²(G) and X is either L^p(G) for 1<p?∞ or C(G), then we show that every complete norm |·| on X that makes translations from (X,|·|) into itself continuous is equivalent to ||·||_p or ||·||_∞ respectively. If 1<p?∞ and every left invariant linear functional on L^p(G) is a constant multiple of the Haar integral, then we show that every complete norm |·| on L^p(G) that makes translations from (L^p(G),|·|) into itself continuous and that makes the map t?L_t from G into bounded is equivalent to ||·||_p.     

Weak solution of (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0))u′(0) = ku(0), r(L, u(L))u′ (L) = hu(L) and k, h are suitable elements of [0, ∞]

We consider weak solutions to the nonlinear boundary value problem (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0)) u′(0) = ku(0), r(L, u(L)) u′(L) = hu(L) and k, h are suitable elements of [0, ∞]. In addition to studying some new boundary conditions, we also relax the constraints on r(x, u) and (Fu)(x). r(x, u) > 0 may have a countable set of jump discontinuities in u and r(x, u)^?1?L_q((0, L) × (0, p)). F is an operator from a suitable set of functions to a subset of L_p(0, L) which have nonnegative values. F includes, among others, examples of the form (Fu)(x) = (1 ? H(x ? x₀)) u(x₀), (Fu)(x) = ∫_x^Lf(y, u(y)) dy where f(y, u) may have a countable set of jump discontinuities in u or F may be chosen so that (Fu)′(x) = ? g(x, u(x)) u′(x) ? q(x) u(x) ? f(x, u(x)) where q is a distributional derivative of an L₂(0, L) function.     

A Parametric Family of Intersective Polynomials with Galois Group A 4

We give an infinite family of intersective polynomials with Galois group A ₄, the alternating group on four letters.     

Equations with absolute Galois group isomorphic to An

Criteria are given for polynomials of the type Xⁿ + aX³ + bX² + cX + d, to have Galois group over any finite number field isomorphic to A_n. We use them to construct, for every n, infinitely many polynomials with absolute Galois group isomorphic to A_n, covering so, the case n even, $\text{4 ? n}$, for which explicit equations were not known.     

Fields with almost small absolute Galois group

We construct and study fields F with the property that F has infinitely many extensions of some fixed degree, but E^×/(E^×)ⁿ is finite for every finite extension E/F and every n ∈ N.