122256711
Exceptional Polynomials of Affine Type
Abstract Let K be a finite field of characteristic p . A polynomial f with coefficients in K is said to be exceptional if it induces a permutation on infinitely many finite extensions of K . Let t be a transcendental, and K be an algebraic closure of K . The exceptional polynomials known to date are constructed from certain twists of polynomials g , such that the splitting field of g ( X ) −  t over K is rational. On the other hand we know that except for p  = 2 or 3, either the degree of an exceptional polynomial f is a power of the characteristic p , or f is a Dickson polynomial. The case deg  f  =  p has been settled completely. We give a new series of exceptional polynomials of degree p m with m even which does not follow the above mentioned construction principle. We show under certain additional hypotheses that the associated monodromy groups of exceptional polynomials will be severely restricted. In particular, we determine precisely what the geometric monodromy groups of exceptional polynomials of degree p 2 will be.
17569071
A Weil-Bound Free Proof of Schur's Conjecture
Letfbe a polynomial with coefficients in the ring OKof integers of a number field. Suppose thatfinduces a permutation on the residue fields OK/p for infinitely many nonzero prime ideals p of OK. Then Schur's conjecture, namely thatfis a composition of linear and Dickson polynomials, has been proved by M. Fried. All the present versions of the proof use Weil's bound on the number of points of absolutely irreducible curves over finite fields in order to get a Galois theoretic translation and to finish the proof by means of finite group theory. This note replaces the use of this deep result by elementary arguments.
123375672
Transitive linear groups with large soluble normal subgroups
131765741
Primitive Monodromy Groups of Polynomials
For a polynomial f ∈ C[X], let G be the Galois group of the Galois closure of the field extension C(X)|C(f(X)). We classify the groups G in the indecomposable case. For polynomials with rational coefficients there are, besides four infinite series, only three more “sporadic” examples. In the Appendix we reprove the classical Theorems of Ritt about decompositions of polynomials using the group-theoretic setup.
5739920
An infinite series of Kronecker conjugate polynomials
Let K be a field of characteristic 0, t a transcendental over K, and Ir be the absolute Galois group of K(t). Then two non-constant polynomials f, g E K[X] are said to be Kronecker conjugate if an element of F fixes a root of f (X) t if and only if it fixes a root of g(X) t. If K is a number field, and f, g E OK [X] where OK is the ring of integers of K, then f and g are Kronecker conjugate if and only if the value set f((OK) equals g(OK) modulo all but finitely many non-zero prime ideals of O9K. In 1968 H. Davenport suggested the study of this latter arithmetic property. The main progress is due to M. Fried, who showed that under certain assumptions the polynomials f and g differ by a linear substitution. Further, he found non-trivial examples where Kronecker conjugacy holds. Until now there were only finitely many known such examples. This paper provides the first infinite series. The main part of the construction is group theoretic.
123589108
On simple semiabelian p-adic lie algebras
118540313
On the diffusion of momentum and mass by internal gravity waves