15736393
Kronecker conjugacy of polynomials
Let f, g ∈ Z[X] be non-constant polynomials with integral coefficients. In 1968 H. Davenport raised the question as to when the value sets f(Z) and g(Z) are the same modulo all but finitely many primes. The main progress until now is M. Fried’s result that f and g then differ by a linear substitution, provided that f is functionally indecomposable. We extend this result to polynomials f of composition length 2. Also, we study the analog when Z is replaced by the integers of a number field. The above number theoretic property translates to an equivalent property of subgroups of a finite group, known as Kronecker conjugacy, a weakening of conjugacy which has been studied by various authors under different assumptions and in other contexts. We also give a simplified and strengthened version of the Galois theoretic translation to finite groups.
123375672
Transitive linear groups with large soluble normal subgroups
119896747
On a Question of Davenport
Letkbe a number field and denote by okits ring of integers. Let p be a non-zero prime ideal of ok. Denote byfthe polynomial derived fromfby reducing the coefficients modulo p. SetVp(f)={f(u)∣u∈ok/p}. Davenport raised the following question (withkbeing the rationals). Supposefandgare polynomials in ok[X] such thatVp(f)=Vp(g) for all but finitely many non-zero prime ideals of ok. Does this implyf(X)=g(aX+b) for somea, b∈k? Extending work of M. Fried, we give an affirmative answer under rather general conditions, and also new types of counter-examples.
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.
59486970
Hilbert’s irreducibility theorem for prime degree and general polynomials
Letf (X, t)εℚ[X, t] be an irreducible polynomial. Hilbert’s irreducibility theorem asserts that there are infinitely manyt0εℤ such thatf (X, t0) is still irreducible. We say thatf (X, t) isgeneral if the Galois group off (X, t) over ℚ(t) is the symmetric group in its natural action. We show that if the degree off with respect toX is a prime ≠ 5 or iff is general of degree ≠ 5, thenf (X, t0) is irreducible for all but finitely manyt0εℤ unless the curve given byf (X, t)=0 has infinitely many points (x0,t0) withx0εℚ,t0εℤ. The proof makes use of Siegel’s theorem about integral points on algebraic curves, and classical results about finite groups, going back to Burnside, Schur, Wielandt, and others.
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.
118540313
On the diffusion of momentum and mass by internal gravity waves