Generic polynomial

About: Generic polynomial is a research topic. Over the lifetime, 608 publications have been published within this topic receiving 6784 citations.

On the Galois group of generalized Laguerre polynomials

Abstract: Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be "large." For a fixed � ∈ Q − Z<0, Filaseta and Lam have shown that the nth degree Generalized Laguerre Polynomial L (�) n (x) = P n=0 n+� n j � (−x) j /j! is irreducible for all large enough n. We use our criterion to show that, under these condi- tions, the Galois group of L (�) n (x) is either the alternating or symmetric group on n letters, generalizing results of Schur for � = 0,1.

Rational connectedness and Galois covers of the projective line

Abstract: Let k be a p-adic field. Some time ago, D. Harbater [9] proved that anyfinite group G may be realized as a regular Galois group over the rationalfunction field in one variable k(t), namely there exists a finite field extensionF/k(t), Galois with group G, such that F is a regular extension of k (i.e. kis algebraically closed in F). Moreover, one may arrange that a given k-placeof k(t) be totally split in F. Harbater proved this theorem for k an arbitrarycomplete valued field. Rather formal arguments ([10, §4.5]; §2 hereafter) thenimply that the theorem holds over any ‘large’ field k. This in turn is a specialcase of a result of Pop [15], hence will be referred to as the Harbater/Poptheorem. We refer to [10], [16], [6] for precise references to the literature (workof D`ebes, Deschamps, Fried, Haran, Harbater, Jarden, Liu, Pop, Serre, andV¨olklein).Most proofs (see [10], [19, 8.4.4, p. 93] and Liu’s contribution to [16]; seehowever [15]) first use direct arguments to establish the theorem when G is acyclic group (here the nature of the ground field is irrelevant), then proceed bypatching, using either formal or rigid geometry, together with GAGA theorems.In the present paper, where I take the case of algebraically closed fieldsfor granted, I show how a technique recently developed by Kolla´r [12] may beused to give a quite different proof of the Harbater/Pop theorem, when the‘large’ field k has characteristic zero. This proof actually gives more than theoriginal result (see comment after statement of Theorem 1).Before I formally state the main result, let us recall what a ‘large’ field is.Let k be a field and let k((y)) be the quotient field of the ring k[[y]] of formalpower series in one variable. Following F. Pop, we shall say that k is ‘large’ ifit satisfies one of the three equivalent properties ([15, Prop. 1.1]):(i) It is existentially closed in k((y)): any k-variety with a k((y))-point hasa k-point.(ii) On a smooth integral k-variety with a k-point, k-points are Zariski dense.(iii) On a smooth integral k-curve with a k-point, k-points are Zariski dense.

Noether's problem for dihedral 2-groups

Abstract: Let K be any field and G be a finite group. Let G act on the rational function field K(xg : g ∈ G )b yK-automorphisms defined by g · xh = xgh for any g, h ∈ G.D enote byK(G) the fixed field K(xg : g ∈ G) G. Noether's problem asks whether K(G) is rational (= purely transcendental) over K. We shall prove that K(G )i s rational overK if G is the dihedral group (resp. quasi-dihedral group, modular group) of order 16. Our result will imply the existence of the generic Galois extension and the existence of the generic polynomial of the corresponding group.