Showing papers on "Generic polynomial published in 2013"

Simple G-graded algebras and their polynomial identities

Abstract: Let G be any group and F an algebraically closed field of characteristic zero. We show that any G-graded finite dimensional associative G-simple algebra over F is determined up to a G-graded isomorphism by its G-graded polynomial identities. This result was proved by Koshlukov and Zaicev in case G is abelian.

Extractors for Polynomial Sources over Fields of Constant Order and Small Characteristic

Abstract: A polynomial source of randomness over F n is a random variable X = f(Z) where f is a polynomial map and Z is a random variable distributed uniformly over F r for some integer r. The three main parameters of interest associated with a polynomial source are the order q of the field, the (total) degree D of the map f , and the base-q logarithm of the size of the range of f over inputs in F r , denoted by k. For simplicity we call X a (q; D; k)-source.

Detecting Fast solvability of equations via small powerful Galois groups

Abstract: Fix an odd prime $p$, and let $F$ be a field containing a primitive $p$th root of unity. It is known that a $p$-rigid field $F$ is characterized by the property that the Galois group $G_F(p)$ of the maximal $p$-extension $F(p)/F$ is a solvable group. We give a new characterization of $p$-rigidity which says that a field $F$ is $p$-rigid precisely when two fundamental canonical quotients of the absolute Galois groups coincide. This condition is further related to analytic $p$-adic groups and to some Galois modules. When $F$ is $p$-rigid, we also show that it is possible to solve for the roots of any irreducible polynomials in $F[X]$ whose splitting field over $F$ has a $p$-power degree via non-nested radicals. We provide new direct proofs for hereditary $p$-rigidity, together with some characterizations for $G_F(p)$ -- including a complete description for such a group and for the action of it on $F(p)$ -- in the case $F$ is $p$-rigid.

On local Galois representations associated to ordinary Hilbert modular forms

Abstract: Let F be a totally real field and p be an odd prime which splits completely in F. We show that a generic p-ordinary non-CM primitive Hilbert modular cuspidal eigenform over F of parallel weight two or more must have a locally non-split p-adic Galois representation, at at least one of the primes of F lying above p. This is proved under some technical assumptions on the global residual Galois representation. We also indicate how to extend our results to nearly ordinary families and forms of non-parallel weight.

The inverse Galois problem for PSL_2(F_p)

Abstract: We show that the simple group PSL_2(F_p) occurs as the Galois group of an extension of the rationals for all primes p>3. We obtain our Galois extensions by studying the Galois action on the second etale cohomology groups of a specific elliptic surface.

Galois realizability of groups of orders p 5 and p 6

Abstract: Let p be an odd prime and k an arbitrary field of characteristic not p. We determine the obstructions for the realizability as Galois groups over k of all groups of orders p 5 and p 6 that have an abelian quotient obtained by factoring out central subgroups of order p or p 2. These obstructions are decomposed as products of p-cyclic algebras, provided that k contains certain roots of unity.

Specialization results and ramification conditions

Abstract: Given a hilbertian field $k$ of characteristic zero and a finite Galois extension $E/k(T)$ with group $G$ such that $E/k$ is regular, we produce some specializations of $E/k(T)$ at points $t_0 \in \mathbb{P}^1(k)$ which have the same Galois group but also specified inertia groups at finitely many given primes. This result has two main applications. Firstly we conjoin it with previous works to obtain Galois extensions of $\mathbb{Q}$ of various finite groups with specified local behavior - ramified or unramified - at finitely many given primes. Secondly, in the case $k$ is a number field, we provide criteria for the extension $E/k(T)$ to satisfy this property: at least one Galois extension $F/k$ of group $G$ is not a specialization of $E/k(T)$.

The fields of definition of branched Galois covers of the projective line

Abstract: In this paper I explore the structure of the fields of definition of Galois branched covers of the projective line over \bar Q. The first main result states that every mere cover model has a unique minimal field of definition where its automorphisms are defined, and goes on to describe special properties of this field. One corollary of this result is that for every G-Galois branched cover there is a field of definition which is Galois over its field of moduli, with Galois group a subgroup of Aut(G). The second main theorem states that the field resulting by adjoining to the field of moduli all of the roots of unity whose order divides some power of |Z(G)| is a field of definition. By combining this result with results from an earlier paper, I prove corollaries related to the Inverse Galois Problem. For example, it allows me to prove that for every finite group G, there is an extension of number fields Q \subset E \subset F such that F/E is G-Galois, and E/Q ramifies only over those primes that divide |G|. I.e., G is realizable over a field that is "close" to Q.

Generic Polynomials for Transitive Permutation Groups of Degree 8 and 9

Abstract: We compute generic polynomials for certain transitive permuta- tion groups of degree 8 and 9, namely SL(2,3), the generalized dihedral group: C2 ⋉ (C3 × C3), and the Iwasawa group of order 16: M16. Rikuna proves the ex- istence of a generic polynomial for SL(2,3) in four parameters in (13); we extend a computation of Grobner in (5) to give an alternative proof of existence for this group's generic polynomial. We establish that the generic dimension and essential dimension of the generalized dihedral group are two. We establish over the rationals that the generic dimension and essential dimension of SL(2,3) and M16 are four.

Existence and Solution by Radicals

Abstract: We consider proofs that every polynomial has one zero (and hence n) in the complex plane This was proved by Gauss in 1799, although a flaw in his proof was pointed out and fixed by Ostrowski in 1920, whereas other scientists had previously made unsuccessful attempts We give details of Gauss’ fourth (trigonometric) proof, and also more modern proofs, such as several based on integration, or on minimization We also treat the proofs that polynomials of degree 5 or more cannot in general be solved in terms of radicals We define groups and fields, the set of congruence classes mod p (x), extension fields, algebraic extensions, permutations, the Galois group We quote the fundamental theorem of Galois theory, the definition of a solvable group, and Galois’ criterion (that a polynomial is solvable by radicals if and only if its group is solvable) We prove that for n ⩾ 5 the group S n is not solvable Finally we mention that a particular quintic has Galois group S 5 , which is not solvable, and so the quintic cannot be solved by radicals

Solvability of Polynomials and Galois Group

Abstract: Solution of polynomial plays fundamental role in the solution of characteristics differential equation to many physical problems. It has been found that Galois Theory can be used to determine the solvability of polynomials over a field by radicals. That is ''if a polynomial is solvable by radicals, then the automorphism group of its splitting field must be a solvable group.'' Field theory is connected with Group theory.

The Galois Endomorphism Ring of a Galois Azumaya Extension

Abstract: Let B be a Galois Azumaya extension of B G with Galois group G; that is, B is a Galois extension of B G with Galois group G which is an Azumaya C G -algebra where C is the center of B. Denote B G by D and the endomorphism ring Hom(DB, DB) of the left D-module endomorphisms of B by Ω. Then Ω is a Galois and a Hirata separable

Periods of polynomials over a Galois ring

Abstract: The period of a monic polynomial over an arbitrary Galois ring GR(pe, d) is theoretically determined by using its classical factorization and Galois extensions of rings For a polynomial f(x) the modulo p remainder of which is a power of an irreducible polynomial over the residue field of the Galois ring, the period of f(x) is characterized by the periods of the irreducible polynomial and an associated polynomial of the form (x−1)m + pg(x) Further results on the periods of such associated polynomials are obtained, in particular, their periods are proved to achieve an upper bound value in most cases As a consequence, the period of a monic polynomial over GR(pe, d) is equal to pe−1 times the period of its modulo p remainder polynomial with a probability close to 1, and an expression of this probability is given