rationality problem of three dimensional monomial group actions abstract let k be a field of characteristic not two and k x y z the rational function field over k with three variables x y z let g be a finite group acting on k x y z by monomial k automorphisms we consider the rationality problem of the fixed field k x y z g under the action of g namely whether k x y z g is rational that is purely transcendental over k or not we may assume that g is a subgroup of gl 3 z and the problem is determined up to conjugacy in gl 3 z there are 73 conjugacy classes of g in gl 3 z by results of endo miyata voskresenski lenstra saltman hajja kang and yamasaki 8 conjugacy classes of 2 groups in gl 3 z have negative answers to the problem under certain monomial actions over some base field k and the necessary and sufficient condition for the rationality of k x y z g over k is given in this paper we show that the fixed field k x y z g under monomial action of g is rational over k except for possibly negative 8 cases of 2 groups and unknown one case of the alternating group of degree four moreover we give explicit transcendental bases of the fixed fields over k for the unknown case we obtain an affirmative solution to the problem under some conditions in particular we show that if k is quadratically closed field then k x y z g is rational over k we also give an application of the result to 4 dimensional linear noether s problem

cyclic length in the tame brauer group of the function field of a p adic curve let f be the function field of a smooth curve over the p adic number field qp we show that for each prime to p number n the n torsion subgroup h 2 f n nbr f is generated by z n cyclic classes in fact the z n length is equal to two it follows that the brauer dimension of f is two first proved in 20 and any f division algebra of period n and index n 2 is decomposable

noether s problem for some semidirect products abstract let k be a field g be a finite group k x g g g be the rational function field with the variables x g where g g the group g acts on k x g g g by k automorphisms where h x g x h g for all h g g let k g be the fixed field defined by k g k x g g g g f k x g g g h f f for all h g noether s problem asks whether the fixed field k g is rational purely transcendental over k let m and n be positive integers and assume that there is an integer t such that t z m z is of order n define a group g m n m n 1 1 t c m c n assume furthermore that i m is an odd integer and ii for any e n the ideal e t m in z e is a principal ideal where e is a primitive e th root of unity theorem if k is a field with m n k then k g m n is rational over k consequently it may be shown that for any positive integer n the set s p p is a prime number such that c g p n is rational over c is of positive dirichlet density in particular s is an infinite set

relative negligibility of linear automorphisms let u and v be finite dimensional vector spaces over a field k gl u gl v and i be the identity transformation on v denote by and i the induced linear automorphisms on u v and i can also be regarded as k automorphisms on the function field k u v it is elementary to check whether and i are conjugate within gl u v by examining their rational canonical forms in this paper we shall give necessary and sufficient conditions for and i to be conjugate within autk k u v for this characterization we introduce the concept of the generalized order through this invariant we also settle the question of when two different polynomials are minimal polynomials of the same linear automorphism of a rational function field

from geometric quantization to moyal quantization it is shown how the moyal product of phase space functions and the weyl correspondence between symbols and operator kernels may be obtained directly using the procedures of geometric quantization applied to the symplectic groupoid constructed by doubling the phase space

rationality of finite groups of monomial automorphisms of k x y abstract it has been shown hajja j algebra 85 1983 243 250 that every finite cyclic group of monomial k automorphisms of k x y has a purely transcendental fixed field here the same result is established with the assumption that the group is cyclic dropped

bogomolov multipliers and retract rationality for semidirect products abstract let g be a finite group the bogomolov multiplier b 0 g is constructed as an obstruction to the rationality of c v g where g g l v is a faithful representation over c we prove that for any finite groups g 1 and g 2 b 0 g 1 g 2 b 0 g 1 b 0 g 2 under the restriction map if g n g 0 with gcd n g 0 1 then b 0 g b 0 n g 0 b 0 g 0 under the restriction map for any integer n we show that there are non direct product p groups g 1 and g 2 such that b 0 g 1 and b 0 g 2 contain subgroups isomorphic to z p z n and z p n z respectively on the other hand if k is an infinite field and g n g 0 where n is an abelian normal subgroup of exponent e satisfying that e k we will prove that if k g 0 is retract k rational then k g is also retract k rational provided that certain local conditions are satisfied this result generalizes previous results of saltman and jambor 18

nature of singularities for analyticity domains of invariant curves we present theoretical arguments based on infinite dimensional bifurcation theory and numerical evidence based on nonperturbative methods that the boundaries of analyticity of invariant curves can be described as an accumulation of branch points which are typically of order 2 we show how this fact would explain previous numerical results of several authors and how it suggests more efficient numerical algorithms which we implement

cyclic galois extensions and normal bases a kummer theory is presented which does not need roots of unity in the ground ring for r commutative with p i e r we study the group of cyclic galois extensions of fixed degree pn in detail our theory is well suited for dealing with cyclic pn extensions of a number field k which are unramified outside p we then consider the group gal 6f p i cpn of all such extensions and its subgroup nb k k p1 cpn of extensions with integral normal basis outside p for the size of the latter we get a simple asymptotic formula n oo and the discrepancy between the two groups is in some way measured by the defect a in leopoldt s conjecture introduction this work begins with a general study of galois extensions of a commutative ring r with finite abelian galois group g part i the results are applicable in a number theoretical setting part ii and give theorems concerning the existence of p integral normal bases in cpn extensions and zp extensions of an arbitrary number field k here p is an odd prime cpn is the cyclic group of order pn and p integral means integral outside places over p let us start by describing the general situation let r be a commutative ring with 1 and g a finite abelian group we study the finite abelian group gal r g which consists of the isomorphism classes of g galois ring extensions of r such extensions were defined by chase harrison and rosenberg 4 the group structure on gal r g is due to harrison 13 and goes back to hasse 14 for the case r k a field in that case the basic idea is to admit some nonfields galois algebras in order to form gal k g the principal link between general galois theory of rings and number theory is the study of ramification let l be a g galois field extension of the number field k x a set of finite places of k and r the ring of x integers in k then the integral closure s of r in l is a g galois extension of r if and only if l k is unramified at all finite places outside x see 4 remark 1 5d in particular 6 l is galois over k iff l k is unramified except at infinity received by the editors june 13 1989 1980 mathematics subject classification 1985 revision primary 1 lr33 1 lr27 13b05

retract rational fields abstract let k be an infinite field the notion of retract k rationality was introduced by saltman in the study of noether s problem and other rationality problems we will investigate the retract rationality of a field in this paper theorem 1 let k k l be fields if k is retract k rational and l is retract k rational then l is retract k rational theorem 2 for any finite group g containing an abelian normal subgroup h such that g h is a cyclic group for any complex representation g gl v the fixed field c v g is retract c rational theorem 3 if g is a finite group then all the sylow subgroups of g are cyclic if and only if c m g is retract c rational for all g lattices m for all short exact sequences 0 c m m 0 because the unramified brauer group of a retract c rational field is trivial theorems 2 and 3 generalize previous results of bogomolov and barge respectively see theorems 5 9 and 6 1

the rationality problem for finite subgroups of gl4 q abstract let g be a finite subgroup of gl 4 q the group g induces an action on q x 1 x 2 x 3 x 4 the rational function field of four variables over q theorem the fixed subfield q x 1 x 2 x 3 x 4 g f q x 1 x 2 x 3 x 4 f f for any g is rational i e purely transcendental over q except for two groups which are images of faithful representations of c 8 and c 3 c 8 into gl 4 q both fixed fields for these two exceptional cases are not rational over q there are precisely 227 such groups in gl 4 q up to conjugation the answers to the rationality problem for most of them were proved by kitayama and yamasaki 2009 ky except for four cases we solve these four cases left unsettled by kitayama and yamasaki thus the whole problem is solved completely

brauer hilbertian fields let f be a field of characteristic p p 0 allowed and let f t be the rational function field in one variable over f we say f is brauer hilbertian if the following holds for every in the brauer group br f t of exponent prime to p there are infinitely many specializations t a f such that the specialization br f is defined and has exponent equal to that of we show every global field is brauer hilbertian and if k is hilbertian and f is finite separable over k t f is brauer hilbertian

negative solutions to three dimensional monomial noether problem three dimensional monomial noether problem can have negative solutions for 8 groups by the suitable choice of the coefficients we find the necessary and sufficient condition for the coefficients to have a negative solution the results are obtained by two criteria of irrationality using galois cohomology

semiclassical analysis of wigner 3j symbol we analyse the asymptotics of the wigner 3j symbol as a matrix element connecting eigenfunctions of a pair of integrable systems obtained by lifting the problem of the addition of angular momenta into the space of schwinger s oscillators a novel element is the appearance of compact lagrangian manifolds that are not tori due to the fact that the observables defining the quantum states are noncommuting these manifolds can be quantized by generalized bohr sommerfeld rules and yield all the correct quantum numbers the geometry of the classical angular momentum vectors emerges in a clear manner efficient methods for computing amplitude determinants in terms of poisson brackets are developed and illustrated

singularities of periodic orbits near invariant curves abstract we show numerically for standard like maps how the singularities in the complex parameter of the function which conjugates the map to a rotation of rational period behave when the period goes to an irrational number furthermore we propose a numerical method to extrapolate the radius of convergence of the series parametrizing the solution of periodic orbits the results are compared with analyses performed by pade approximants greene s method root criterion and the prediction by renormalization theory

generic extensions and generic polynomials for multiplicative groups let a a be a finite dimensional algebra over a finite field f q f q and let g a g a be the multiplicative group of a a in this paper we construct explicitly a generic galois g extension s r s r where r is a localized polynomial ring over f q f q and an explicit generic polynomial for g in dim f q a dim f q a parameters

degree three unramified cohomology groups abstract let k be any field g be a finite group let g act on the rational function field k x g g g by k automorphisms defined by h x g x h g for any g h g denote by k g k x g g g g the fixed subfield noether s problem asks whether k g is rational purely transcendental over k the unramified brauer group br nr c g and the unramified cohomology h nr 3 c g q z are obstructions to the rationality of c g see 14 and 5 peyre proves that if p is an odd prime number then there is a group g such that g p 12 br nr c g 0 but h nr 3 c g q z 0 thus c g is not stably c rational 12 using peyre s method we are able to find groups g with g p 9 where p is an odd prime number such that br nr c g 0 h nr 3 c g q z 0

on noether s problem for central extensions of symmetric and alternating groups for a field k and a finite group g acting regularly on a set of indeterminates x xg g g let k g denote the invariant field k x g we first prove for the alternating group an that if n is odd then q an is rational over q an 1 we then obtain an analogous result where an is replaced by an arbitrary finite central extension of either an or sn valid over q n for suitable n concrete applications of our results yield 1 a new proof of maeda s result on the rationality of q x1 x5 a5 q 2 an affirmative answer to noether s problem over q for both a5 and s5 3 an affirmative answer to noether s problem over c for every finite central extension group of either an or sn with n 5

generic extensions and generic polynomials for linear algebraic groups abstract we construct generic extensions and polynomials for certain linear algebraic groups over finite fields in particular we show the existence of generic polynomials for the cyclic group c 2 m in characteristic p for all odd primes p

natural boundaries for area preserving twist maps we consider kam invariant curves for generalizations of the standard map of the form x y x y y f x wheref x is an odd trigonometric polynomial we study numerically their analytic properties by a pade approximant method applied to the function which conjugates the dynamics to a rotation in the complex plane natural boundaries of different shapes are found in the complex plane the analyticity region appears to be a strip bounded by a natural boundary whose width tends linearly to 0 as tends to the critical value

semiclassical mechanics of the wigner 6j symbol the semiclassical mechanics of the wigner 6j symbol is examined from the standpoint of wkb theory for multidimensional integrable systems to explore the geometrical issues surrounding the ponzano regge formula the relations among the methods of roberts and others for deriving the ponzano regge formula are discussed and a new approach based on the recoupling of four angular momenta is presented a generalization of the yutsis type of spin network is developed for this purpose special attention is devoted to symplectic reduction the reduced phase space of the 6j symbol the 2 sphere of kapovich and millson and the reduction of poisson bracket expressions for semiclassical amplitudes general principles for the semiclassical study of arbitrary spin networks are laid down some of these were used in our recent derivation of the asymptotic formula for the wigner 9j symbol

computation of domains of analyticity for some perturbative expansions of mechanics abstract we compute the domain of analyticity of some perturbative expansions for invariant circles appearing in mechanics we use pade approximants for the perturbative expansions and introduce methods to ascertain their domain of convergence we also use non perturbative methods based on direct computation of the invariant circles and in analogy with greene s criterion approximation by circles with rational rotation we find that the domains computed by all the methods agree within the limits of accuracy we also study rigorously the nature of the singularities when the frequency is rational

asymptotics of clebsch gordan coefficients asymptotic expressions for clebsch gordan coefficients are derived from an exact integral representation both the classically allowed and forbidden regions are analyzed higher order approximations are calculated these give for example six digit accuracy when the quantum numbers are in the hundreds

rationality of p group actions abstract let p be a prime number g a p group of order p 4 k a field with char k p if the exponent of g is p e and k contains a primitive p e th root of unity then k v g is rational purely transcendental over k where g gl v is any linear representation of g over k

noether s problem for p groups with a cyclic subgroup of index p2 abstract let k be any field and g be a finite group let g act on the rational function field k x g g g by k automorphisms defined by g x h x g h for any g h g noether s problem asks whether the fixed field k g k x g g g g is rational purely transcendental over k we will prove that if g is a non abelian p group of order p n n 3 containing a cyclic subgroup of index p 2 and k is any field containing a primitive p n 2 th root of unity then k g is rational over k as a corollary if g is a non abelian p group of order p 3 and k is a field containing a primitive p th root of unity then k g is rational

noether s problem on semidirect product groups abstract let k be a field g a finite group let g act on the function field l k x g by x x for any g denote the fixed field of the action by k g l g f g l f g f g g noether s problem asks whether k g is rational purely transcendental over k it is known that if g c m c n is a semidirect product of cyclic groups c m and c n with z n a unique factorization domain and k contains an eth primitive root of unity where e is the exponent of g then k g is rational over k in this paper we give another criteria to determine whether k c m c n is rational over k in particular if p q are prime numbers and there exists x z q such that the norm n q q q x p then c c p c q is rational over c

division algebras over surfaces abstract we consider here a number of topics concerning the theory of division algebras over the function field of a surface one result relates the obstruction for ramification data to be from a division algebra and third etale cohomology another result shows this obstruction is always zero when the surface is spec of a regular local ring with some mild assumptions at the same time we study the brauer group of this function field as it relates to the brauer group of the function field of the henselization finally we prove a result which says that brauer group elements which look like they are of prime index q unequal to any characteristic must have all their ramification split by a cyclic galois extension of the same degree this last result requires a primitive q root of one

a note on monomial automorphisms abstract previous results hajja j algebra73 1981 30 36 on monomial automorphisms are strengthened and the rationality of all monomial automorphisms on k x y is established the relation of certain steps in the proofs to a special case of a problem of zariski on whether quasi rationality implies rationality is also noted

quasi monomial actions and some 4 dimensional rationality problems abstract let g be a finite group acting on k x 1 x n the rational function field of n variables over a field k the action is called a purely monomial action if x j 1 i n x i a i j for all g for 1 j n where a i j 1 i j n gl n z the main question is that under what situations the fixed field k x 1 x n g is rational purely transcendental over k this rationality problem has been studied by hajja kang hoshi rikuna when n 3 in this paper we will prove that k x 1 x 2 x 3 x 4 g is rational over k provided that the purely monomial action is decomposable to prove this result we introduce a new notion the quasi monomial action which is a generalization of previous notions of multiplicative group actions moreover we determine the rationality problem of purely quasi monomial actions of k x y g over k where k k g

generic picard vessiot extensions for connected by finite groups abstract we construct generic picard vessiot extensions for linear algebraic groups g which are isomorphic to the semidirect product of a connected group g 0 by a finite group h where the adjoint h action on the lie algebra of g 0 is faithful

deformations of dihedral 2 group extensions of fields given a g galois extension of number fields l k we ask whether it is a specialization of a regular g galois cover of pk this is the inverse of the usual use of the hilbert irreducibility theorem in the inverse galois problem we show that for many groups such arithmetic liftings exist by observing that the existence of generic extensions implies the arithmetic lifting property we explicitly construct generic extensions for dihedral 2 groups under certain assumptions on the base field k we also show that dihedral groups of order 8 and 16 have generic extensions over any base field k with characteristic different from 2

negligibility of projective linear automorphisms abstract letkbe any field s aij 1 i j n 1 gln 1 k k k x1 xn 1 xn xn 1 y1 yn 1 yn yn 1 a field extension of transcendence degree 2noverk the diagonal action ofsonkis defined as equation equation theorem there existz1 zn ksuch thatk k x1 xn 1 xn xn 1 z1 zn ands zi zifor 1 i n in particular ifk x1 xn 1 xn xn 1 s is rational purely transcendental overk so isk s overk

noether s problem for four and five dimensional linear actions abstract let g be a finite group k be a field and g gl v be a faithful representation where v is a finite dimensional vector space over k in this paper we will consider the problem which asks whether the fixed field k v g is rational i e purely transcendental over k this is a variant of noether s problem we will show the complete results of this problem in the case where g is a 2 group k q and dim v 4 and 5 our results imply the existence of a generic polynomial for the corresponding group

some actions of symmetrical groups abstract let k be any field v the n dimensional vector space over k on which the symmetric group sn acts by permuting a basis then the fixed fields k skv sn and k k v sn are rational over k where skv and k v are the kth symmetric power and exterior power of v respectively the rationality of some other fixed fields are established also

description of galois unipotent extensions abstract given an arbitrary field f we describe all galois extensions l f whose galois groups are isomorphic to the group of upper triangular unipotent 4 by 4 matrices with entries in the field of two elements

rationality of three dimensional quotients by monomial actions let g be a finite 2 group and k be a field satisfying that i chark 2 and ii a k for any a k if g acts on the rational function field k x y z by monomial k automorphisms then the fixed field k x y z g is rational purely transcendental over k applications of this theorem will be given

quantum normal forms moyal star product and bohr sommerfeld approximation a normal form transformation is carried out on one dimensional quantum hamiltonians that transforms them into functions of the quantum harmonic oscillator the method works with the weyl transform or symbol of the hamiltonian the moyal star product is used to carry out the normal form transformation at the level of symbols diagrammatic techniques are developed for handling the expressions that result from higher order terms in the moyal series once the normal form is achieved the bohr sommerfeld formula for the eigenvalues including higher order corrections follows easily

rational invariants for subgroups of s5 and s7 abstract let g be a subgroup of s n the symmetric group of degree n for any field k g acts naturally on the rational function field k x 1 x 2 x n via k automorphisms defined by x i x i for any g any 1 i n theorem if n 5 then the fixed field k x 1 x n g is purely transcendental over k we will show that c x 1 x 7 g is also purely transcendental over c if g is any transitive subgroups of s 7 other than a 7 a similar result is valid for solvable transitive subgroups of s 11

twisted actions of symmetric groups abstract let k be any field and the symmetric group s n act on the rational function field k x 1 x n by x i x i if a n and x i x i 1 if a n we shall show that the fixed field k x 1 x n s n is rational over k similar results for actions of s n on the symmetric powers and exterior powers of v n i 1 k x i are valid

generic extensions and generic polynomials for semidirect products this paper presents a generalization of a theorem of saltman on the existence of generic extensions with group a g over an infinite field k where a is abelian using less restrictive requirements on a and g the method is constructive thereby allowing the explicit construction of generic polynomials for those groups and it gives new bounds on the generic dimension generic polynomials for several small groups are constructed

division algebras with psl 2 q galois maximal subfields abstract if g is a finite group and k is a field then g is k admissible if there exists a g galois extension l k such that l is a maximal subfield of a k division algebra we prove that psl 2 7 is k admissible for any number field which either fails to contain 1 or which has two primes lying over the dyadic prime in addition psl 2 11 is shown to be admissible over q or any number field k with at least two extensions of the dyadic prime indeed there exist infinitely many linearly disjoint admissible extensions for these groups

asymptotics of radial wave equations the langer modification is an improvement in the wkb analysis of the radial schrodinger equation we derive a generalization of the langer modification to any radial operator for differential operators we write the modified classical symbols explicitly and show that the wkb wavefunctions with the modification have the exact limiting behavior for small radius unlike in the schrodinger case generally the modified radial analysis is not equivalent to the wkb analysis of the full problem before reduction by the spherical symmetry

noether s problem for groups of order 32 abstract let k be any field g be a finite group let g act on the rational function field k x g g g by k automorphisms defined by h x g x h g for any g h g denote by k g k x g g g g the fixed field noether s problem asks under what situations the fixed field k g will be rational purely transcendental over k theorem let g be a finite group of order 32 with exponent e if char k 2 or k is any field containing a primitive eth root of unity then k g is rational over k

noether s problem for groups of order 243 abstract let k be any field g be a finite group let g act on the rational function field k x g g g by k automorphisms defined by h x g x h g for any g h g denote by k g k x g g g g the fixed field noether s problem asks under what situations the fixed field k g will be rational purely transcendental over k according to the data base of gap there are 10 isoclinism families for groups of order 243 it is known that there are precisely 3 groups g of order 243 they consist of the isoclinism family 10 such that the unramified brauer group of c g over c is non trivial thus c g is not rational over c we will prove that if 9 k then k g is rational over k for groups of order 243 other than these 3 groups except possibly for groups belonging to the isoclinism family 7

rationality problem for transitive subgroups of s8 abstract for any field k and any transitive subgroup g of s 8 let g act naturally on k x 1 x 8 by permuting the variables we prove that under some minor conditions k x 1 x 8 g is always k rational except g is a 8 or g is isomorphic to pgl 2 7 we pay special attentions on the characteristic 2 cases

frobenius groups and retract rationality abstract let k be any field g be a finite group acting on the rational function field k x g g g by h x g x h g for any h g g define k g k x g g g g noether s problem asks whether k g is rational purely transcendental over k a weaker notion retract rationality introduced by saltman is also very useful for the study of noether s problem we prove that if g is a frobenius group with abelian frobenius kernel then k g is retract k rational for any field k satisfying some mild conditions as an application we show that for any algebraic number field k for any frobenius group g with frobenius complement isomorphic to s l 2 f 5 there is a galois extension field k over k whose galois group is isomorphic to g i e the inverse galois problem is valid for the pair g k the same result is true for any non solvable frobenius group if k 8 is a cyclic extension of k

on generic polynomials abstract we examine over arbitrary fields the possible implications among the concepts due to d saltman of generic galois extension retract rational extension the lifting property for galois extensions the notions due to g smith of generic polynomial and of descent generic polynomial due to f demeyer

twisted multiplicative field invariants noether s problem and galois extensions noether s problem concerns fields f xg 1 g e g g f g where f is a field g is a finite group g acts by permuting the indeterminants x in the obvious way and f g refers to the field of invariant elements in showing the f g s were nonrational for certain g the author was led to consider multiplicative invariant fields ss which are defined as follows let g be a finite group and q a g lattice that is q is a finitely generated z g module which is free as an abelian group form the group algebra f g and its field of fractions f q g acts naturally on f q and the multiplicative invariant field is f q in ss it was shown that for certain q f q g was essentially equivalent to actually stably isomorphic to f g for g a split extension of g with abelian kernel this leads one to ask whether f g can be expressed as a kind of invariant field for g when g is a nonsplit extension of g the answer is yes and is the core of this paper in fact f g will be shown to be a so called a twisted multiplicative invariant field of g the goal of this paper is to present this fact and then draw a series of conclusions using it such a twisted invariant fields have appeared explicitly in previous works in ss these fields arise as another way of describing the invariant fields of reductive algebraic groups note that in s6 the finite group is the weyl group and the lattice is derived from the root lattice in s7 the unramified brauer groups of a twisted invariant fields are calculated

the center of the generic g crossed product abstract let g be a finite group and let f be a field of characteristic zero in this paper we construct a generic g crossed product over f using generic graded matrices the center of this generic g crossed product denoted by f g is then the invariant field of a suitable g action on a field of rational functions in several indeterminates the main goal of this paper is to study the extensions f g f given that f contains enough roots of unity and determine how close they are to being purely transcendental in particular we show that f g f is a stably rational extension for g c 2 c 2 n where n is odd and for g n 2 m e 1 1 where g c d n 2 m 1 furthermore we prove that if h k are groups of coprime orders then f h k is stably rationally equivalent to the fraction field of f h f k

noether s problem and mathbb q generic polynomials for the normalizer of the 8 cycle in s_8 and its subgroups we study noether s problem for various subgroups h of the normalizer ot a group c 8 generated by an 8 cycle in s 8 the symmetric group of degree 8 in three aspects according to the way they act on rational function fields i e q x 0 x 7 q x 1 x 4 and q x y we prove that it has affirmative answers for those h containing c 8 properly and derive a q generic polynomial with four parameters for each h on the other hand it is known in connection to the negative answer to the same problem for c 8 q that there does not exist a q generic polynomial for c 8 this leads us to the question whether and how one can describe for a given field k of characteristic zero the set of c 8 extensions l k one of the main results of this paper gives an answer to this question

the iteration of cubic polynomials part i the global topology of parameter space fonctions univalentes en dynamique analytique complexe torsion de la structure complexe topologie globale de l espace parametre

rationality problem of gl4 group actions abstract let k be any field which may not be algebraically closed v be a four dimensional vector space over k gl v where the order of may be finite or infinite f t k t be the characteristic polynomial of let 1 2 3 be the four roots of f t 0 in some extension field of k theorem 1 bothk v and k p v are rational purelytranscendental overkif at least one of the following conditions is satisfied i char k 2 ii f t is a reducible or inseparable polynomial ink t iii not all of 1 2 3are roots of unity iv iff t is separable irreducible then the galois group off t overkis not isomorphic to the dihedral group of order 8 or the klein four group theorem 2 suppose that all iare roots of unity andf t k t is separable irreducible a if the galois group off t is isomorphic to the dihedral group of order 8 then bothk v and k p v are not stably rational overk b when the galois group off t is isomorphic to the klein four group then a necessary and sufficient condition for rationality ofk v and k p v is provided see theorem 1 5 for details

a rigorous partial justification of greene s criterion we prove several theorems that lend support to greene s criterion for the existence or not of invariant circles in twist maps in particular we show that some of the implications of the criterion are correct when the aubry mather sets are smooth invariant circles or uniformly hyperbolic we also suggest a simple modification that can work in the case that the aubry mather sets have nonzero lyapunov exponents the latter is based on a closing lemma for sets with nonzero lyapunov exponents which may have several other applications

noether s problem for metacyclic p groups abstract let k be any field and g be a finite group let g act on the rational function field k x g g g by k automorphisms defined by g x h x gh for any g h g denote by k g the fixed field k x g g g g noether s problem asks whether k g is rational purely transcendental over k an affirmative answer to noether s problem for metacyclic p groups will be proved provided that k contains enough roots of unity

birational classification of fields of invariants for groups of order 128 let g be a finite group acting on the rational function field c xg g g c xg g g by cc automorphisms h xg xhgh xg xhg for any g h gg h g noether s problem asks whether the invariant field c g k xg g g gc g k xg g g g is rational i e purely transcendental over cc by fischer s theorem c g c g is rational over cc when g is a finite abelian group saltman and bogomolov respectively showed that for any prime p there exist groups g of order p9p9 and of order p6p6 such that c g c g is not rational over cc by showing the non vanishing of the unramified brauer group brnr c g 0brnr c g 0 which is an avatar of the birational invariant h3 x z torsh3 x z tors given by artin and mumford where x is a smooth projective complex variety whose function field is c g c g for p 2p 2 chu hu kang and prokhorov proved that if g is a 2 group of order 32 then c g c g is rational over cc chu hu kang and kunyavskii showed that if g is of order 64 then c g c g is rational over cc except for the groups g belonging to the two isoclinism families 13 13 with brnr c g 0brnr c g 0 and 16 16 with brnr c g c2brnr c g c2 bogomolov and bohning s theorem claims that if g1g1 and g2g2 belong to the same isoclinism family then c g1 c g1 and c g2 c g2 are stably cc isomorphic we investigate the birational classification of c g c g for groups g of order 128 with brnr c g 0brnr c g 0 moravec showed that there exist exactly 220 groups g of order 128 with brnr c g 0brnr c g 0 forming 11 isoclinism families j j we show that if g1g1 and g2g2 belong to 16 16 31 31 37 37 39 39 43 43 58 58 60 60 or 80 80 resp 106 106 or 114 114 then c g1 c g1 and c g2 c g2 are stably cc isomorphic with brnr c gi c2brnr c gi c2 explicit structures of non rational fields c g c g are given for each cases including also the case 30 30 with brnr c g c2 c2brnr c g c2 c2

galois cohomology of fields without roots of unity there is a standard correspondence between elements of the cohomology group h1 f n with the trivial action of f gal fs f on n and cyclic extensions of dimension n over f we extend this to a correspondence between the cohomology groups h1 f n where the action of f on n varies and the extensions of dimension n of k which are galois over f where k f n and k f is prime to n the cohomology groups are also related to eigenspaces of h1 k z n with respect to the natural action of gal k f as a result we extend albert s cyclicity criterion stated in the 1930s for division algebras of prime degree to algebras of prime power degree over f under the assumption stated above we also extend the rosset tate result on the corestriction of cyclic algebras in the presence of roots of unity to extensions in which roots of unity live in an extension of dimension 3 over the base field in particular if roots of unity live in a quadratic extension of the base field then corestriction of a cyclic algebra along a quadratic extension is similar to a product of two cyclic algebras another application is that f central algebras which are split by a certain semidirect product extension of f are cyclic in particular if k f 2 then algebras over f which are split by an odd order dihedral extension are cyclic we also construct generic examples of algebras which become cyclic after extending scalars by roots of unity and show the existence of elements for which most powers have reduced trace zero

complex analytic dynamics on the riemann sphere dichotomie dynamique de fatou et julia points periodiques consequences du theoreme de montel l ensemble de julia est la fermeture de l ensemble des points periodiques repulseurs resultats classiques sur l ensemble de fatou classification de sullivan de l ensemble de fatou une condition pour le developpement sur l ensemble de julia la dynamique des polynomes l ensemble de mandelbrot et le travail de douady et hubbard le theoreme d application de riemann mesurable et la dynamique analytique

numerical calculation of domains of analyticity for perturbation theories in the presence of small divisors we study numerically the complex domains of validity for kam theory in generalized standard mappings we compare methods based on pade approximants and methods based on the study of periodic orbits

rationality of some projective linear actions let k be any field which may not be algebraically closed v a finite dimensional vector space over k gl v where the order of can be finite or infinite theorem if dimk v 3 then both k v and k p v are rational purely transcendental over k similar results hold for a cyclic affine action

on asymptotic expansions of twisted products the series development of the quantum mechanical twisted product is studied the series is shown to make sense as a moment asymptotic expansion of the integral formula for the twisted product either pointwise or in the distributional sense depending on the nature of the factors a condition is given that ensures convergence and is stronger than previously known results possible applications are examined

normalized brauer factor sets abstract we investigate normalized brauer factor sets of central simple algebras with respect to arbitrary maximal separable subalgebras and show that they have a cohomological description as a consequence a central simple algebra of even degree having a normalized brauer factor set cannot be a division algebra an intrinsic equivalent condition is given for a central simple algebra to have a normalized brauer factor set consequently an algebra has a normalized brauer factor set if it is a square in the relative brauer group the converse holds for index 4 or for symbols but an example is given of an algebra of index 8 with normalized brauer factor set which isnota square in the relative brauer group on the other hand supposedis a division algebra of odd degree ifdhas a maximal separable subfieldkwhose galois groupgsatisfies a certain property which automatically holds for g odd thendcontains an elementafor which tr a tr a2 tr a 1 0