Finite Field Extension Research Articles

Norm Principles for Forms of Higher Degree Permitting Composition

Let F be a field of characteristic 0 or greater than d. Scharlau's norm principle holds for finite field extensions K over F, for certain forms ϕ of degree d over F which permit composition.

Difference fields and descent in algebraic dynamics. II

AbstractThis second part of the paper strengthens the descent theory described in the first part to rational maps and arbitrary base fields. We obtain in particular a decomposition of any difference field extension into a tower of finite, field-internal and one-based difference field extensions. This is needed in order to obtain the ‘dynamical Northcott’ Theorem 1.11 of Part I in sharp form.

Efficient pth root computations in finite fields of characteristic p

We present a method for computing pth roots using a polynomial basis over finite fields $${\mathbb F_q}$$ of odd characteristic p, p ? 5, by taking advantage of a binomial reduction polynomial. For a finite field extension $${\mathbb F_{q^m}}$$ of $${\mathbb F_q}$$ our method requires p ? 1 scalar multiplications of elements in $${\mathbb F_{q^m}}$$ by elements in $${\mathbb F_q}$$ . In addition, our method requires at most $${(p-1)\lceil m/p \rceil}$$ additions in the extension field. In certain cases, these additions are not required. If z is a root of the irreducible reduction polynomial, then the number of terms in the polynomial basis expansion of z 1/p , defined as the Hamming weight of z 1/p or $${{\rm wt}\left(z^{1/p} \right)}$$ , is directly related to the computational cost of the pth root computation. Using trinomials in characteristic 3, Ahmadi et al. (Discrete Appl Math 155:260---270, 2007) give $${{\rm wt}\left(z^{1/3} \right)}$$ is greater than 1 in nearly all cases. Using a binomial reduction polynomial over odd characteristic p, p ? 5, we find $${{\rm wt}\left(z^{1/p}\right) = 1}$$ always.

Elliptic periods for finite fields

We construct two new families of basis for finite field extensions. Bases in the first family, the so-called elliptic bases, are not quite normal bases, but they allow very fast Frobenius exponentiation while preserving sparse multiplication formulas. Bases in the second family, the so-called normal elliptic bases are normal bases and allow fast (quasi-linear) arithmetic. We prove that all extensions admit models of this kind.

On the algebra of some group schemes

The algebra of a finite group over a field k of characteristic zero is known to be a projective separable k-algebra; but these separable algebras are of a very special type, characterized by Brauer and Witt. In contrast with that, we prove that any projective separable k-algebra is a quotient of the group algebra of a suitable group scheme, finite etale over k. In particular, any finite separable field extension K⊂L, even a noncyclotomic one, may be generated by a finite etale K-group scheme.

DIVISIBILITY OF EXPONENTIAL SUMS AND SOLVABILITY OF CERTAIN EQUATIONS OVER FINITE FIELDS

In [4], Carlitz determined conditions under which infinite families of polynomials have solutions in a finite field. In this paper we extend some of Carlitz’s results by computing the exact p-divisibility of certain exponential sums. As a by-product we obtain an upper bound for the Waring number for polynomials over extensions of finite fields. ∗franciscastr@gmail.com, Tel: (787)764-0000, X4696, Fax: (787)281-0651 †iverubio@uprrp.edu, Tel: (787)764-0000 (4759), Fax: (787)773-1717 ‡jmanuel jmvg@yahoo.com, Tel: 787-738-2161

The zeros of Dedekind zeta functions and class numbers of CM-fields

Let F'/F be a finite normal extension of number fields with Galois group Gal(F'/F). Let Χ be an irreducible character of Gal(F'/F) of degree greater than one and L(s, x) the associated Artin L-function. Assuming the truth of Artin's conjecture, we have explicitly determined a zero-free region about 1 for L(s,x). As an application we show that, for a CM-field K of degree 2n with solvable normal closure over Q, if n > 370 as well as n ∉ {384,400,416,448,512}, then the relative class number of K is greater than one.

Computing generators of free modules over orders in group algebras

Let E be a number field and G be a finite group. Let A be any O E -order of full rank in the group algebra E [ G ] and X be a (left) A -lattice. We give a necessary and sufficient condition for X to be free of given rank d over A . In the case that the Wedderburn decomposition E [ G ] ≅ ⊕ χ M χ is explicitly computable and each M χ is in fact a matrix ring over a field, this leads to an algorithm that either gives elements α 1 , … , α d ∈ X such that X = A α 1 ⊕ ⋯ ⊕ A α d or determines that no such elements exist. Let L / K be a finite Galois extension of number fields with Galois group G such that E is a subfield of K and put d = [ K : E ] . The algorithm can be applied to certain Galois modules that arise naturally in this situation. For example, one can take X to be O L , the ring of algebraic integers of L, and A to be the associated order A ( E [ G ] ; O L ) ⊆ E [ G ] . The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when K = E = Q .

Pólya and Newtonian function fields

Let \(K/\mathbb {F}_q (T)\) be a finite function field extension and denote by OK the integral closure of \(\mathbb {F}_q [T]\) in K. In this article, we are interested in Polya fields, that is, fields K, such that the OK-module Int(OK) of integer-valued polynomials over OK admits a regular basis. We show that the cyclotomic extensions of \(\mathbb {F}_q (T)\) are Polya fields, and we characterize some totally imaginary extensions which are Polya fields. Then, we are interested in Polya fields K which have a regular basis of the form \(\left\{\prod_{0\le k < n}\frac{X-a_k}{a_n-a_k},\,n\in\mathbb {N}\right\}\) for some sequences \((a_n)_{n\in\mathbb {N}}\) of elements of OK. For totally imaginary extensions, we show that it is the case if and only if OK is isomorphic to \(\mathbb {F}_q [T]\). This gives a answer to a question raised by Thakur.

FINITE CYCLIC TAME EXTENSIONS OF kp((t))

Let p be a prime number and let ℚ/ℤ′ be the elements in ℚ/ℤ of order prime to p. Let [Formula: see text], where c is a prime power of p. We use characters and valuation theory to prove that Δ is a parameter space for the cyclic tame extensions of the formal Laurent series field kp((t)) of degree prime to p. Furthermore, we construct the cyclic tame extension corresponding to a given triple in Δ. The structure of finite cyclic tame extensions of the p-adic number fields was thoroughly investigated by A. A. Albert in 1935. Here we get the same result as consequence of our main theorem.

Multiplicative invariant lattices in [formula omitted] obtained by twisting of group algebras and some explicit characterizations

Let G be a finite group and R G be its group algebra defined over R . If we define in G a 2-cochain F, then we can consider the algebra R F G which is obtained from R G deforming the product, x . F y = F ( x , y ) x y , ∀ x , y ∈ G . Examples of R F ( Z 2 ) n algebras are Clifford algebras and Cayley algebras like octonions. In this paper we consider generalizations of lattices with complex multiplication in the context of these twisted group algebras. We explain how these induce the natural algebraic structure to endow any arbitrary finite-dimensional lattice whose real components stem from any finite algebraic field extension over Q with a multiplicative closed structure. Furthermore, we develop some fully explicit characterizations in terms of generalized trace and norm functions.

Structure of the Brauer ring of a field extension

In 1986, Jacobson has defined the Brauer ring $B(E, D)$ for a finite Galois field extension $E/D$, whose unit group canonically contains the Brauer group of $D$. In 1993, Cheng Xiang Chen determined the structure of the Brauer ring in the case where the extension is trivial. He revealed that if the Galois group $G$ is trivial, the Brauer ring of the trivial extension $E/E$ becomes naturally isomorphic to the group ring of the Brauer group of $E$. In this paper, we generalize this result to any finite group $G$ via the theory of the restriction functor, by means of the well-understood functor $−_+$. More generally, we determine the structure of the $F$-Burnside ring for any additive functor $F$. We construct a certain natural isomorphism of Green functors, which induces the above result with an appropriate $F$ related to the Brauer group. This isomorphism will enable us to calculate Brauer rings for some extensions. We illustrate how this isomorphism provides Green-functor-theoretic meanings for the properties of the Brauer ring shown by Jacobson, and compute the Brauer ring of the extension $ℂ/ℝ$.

Polynomial Imaginary Decompositions for Finite Separable Extensions

Let K be a field and let L = K[ξ] be a finite field extension of K of degree m > 1. If f ∈ L[Z] is a polynomial, then there exist unique polynomials u0, . . . , um−1 ∈ K[X0, . . . , Xm−1] such that f( Pm−1 j=0 ξ Xj) = Pm−1 j=0 ξ uj . A. Nowicki and S. Spodzieja proved that, if K is a field of characteristic zero and f 6= 0, then u0, . . . , um−1 have no common divisor in K[X0, . . . , Xm−1] of positive degree. We extend this result to the case when L is a separable extension of a field K of arbitrary characteristic. We also show that the same is true for a formal power series in several variables.

ORBITAL INTEGRALS FOR LINEAR GROUPS

For a linear group G acting on an absolutely irreducible variety X over Q, we describe the orbits of X(Qp) under G(Qp) and of X(Fp(( t))) under G(Fp(( t))) for p big enough. This allows us to show that the degree of a wide class of orbital integrals over Qp or Fp(( t)) is less than or equal to 0 for p big enough, and similarly for all finite field extensions of Qp and Fp(( t)).

On the tensor rank of the multiplication in the finite fields

First, we prove the existence of certain types of non-special divisors of degree g − 1 in the algebraic function fields of genus g defined over F q . Then, it enables us to obtain upper bounds of the tensor rank of the multiplication in any extension of quadratic finite fields F q by using Shimura and modular curves defined over F q . From the preceding results, we obtain upper bounds of the tensor rank of the multiplication in any extension of certain non-quadratic finite fields F q , notably in the case of F 2 . These upper bounds attain the best asymptotic upper bounds of Shparlinski–Tsfasman–Vladut [I.E. Shparlinski, M.A. Tsfasman, S.G. Vladut, Curves with many points and multiplication in finite fields, in: Lecture Notes in Math., vol. 1518, Springer-Verlag, Berlin, 1992, pp. 145–169].

Congruences between derivatives of abelian L-functions at s=0

Let K/k be a finite abelian extension of global fields. We prove that a natural equivariant leading term conjecture implies a family of explicit congruence relations between the values at s=0 of derivatives of the Dirichlet L-functions associated to K/k. We also show that these congruences provide a universal approach to the ‘refined abelian Stark conjectures’ formulated by, inter alia, Stark, Gross, Rubin, Popescu and Tate. We thereby obtain the first proofs of, amongst other things, the Rubin–Stark conjecture and the ‘refined class number formulas’ of both Gross and Tate for all extensions K/k in which K is either an abelian extension of ℚ or is a function field.

Hopf–Galois structures on almost cyclic field extensions of 2-power degree

Let L / K be a finite separable field extension, and let E be the normal closure of L / K . Let G = Gal ( E / K ) and G ′ = Gal ( E / L ) . We call L / K almost cyclic if G ′ has a normal cyclic complement in G. This includes the case that L / K is a cyclic Galois extension or a radical extension. We give a method for counting Hopf–Galois structures on an almost cyclic extension L / K . We then count the Hopf–Galois structures on an almost cyclic extension of degree 2 n , n ⩾ 3 , and determine how many of them are almost classical. This is analogous to a result of T. Kohl [T. Kohl, Classification of the Hopf–Galois structures on prime power radical extensions, J. Algebra 207 (1998) 525–546] which counts the Hopf–Galois structures on a radical extension of odd prime-power degree. In contrast to the odd prime-power degree case, however, we find that an almost cyclic extension L / K of 2-power degree has Hopf–Galois structures for which the Hopf algebra acting on L is not commutative.

Completeness and compactness for varieties over a local field

This paper shows that for a local field K, a subfield k ⊂ K and a variety X over k, X is complete if and only if for every finite field extension Kʹ | K, the set X(Kʹ) is compact in its strong topology.

Constructing Finite Field Extensions with Large Order Elements

In this paper, we present an algorithm that, given a fixed prime power q and a positive integer N, finds an integer $n \in [N, 2qN]$ and an element $\alpha \in \mbox{\bf F}_{q^n}$ of order greater than $ 5.8^{n / \log_q n}$, in time polynomial in N. We present another algorithm that finds an integer $n \in [N, N+O(N^{0.77})]$ and an element $\alpha \in \mbox{\bf F}_{q^n}$ of order at least $ 5.8^{\sqrt{n}}$, in time polynomial in N. Our result is inspired by the recent AKS primality testing algorithm [M. Agrawal, N. Kayal, and N. Saxena, Ann. of Math. (2), 160 (2004), pp. 781–793] and the subsequent improvements [P. Berrizbeitia, Math. Comp., 74 (2005), pp. 2043–2059, Q. Cheng, in Proceedings of the 23rd Annual International Cryptology Conference (CRYPTO 2003), D. Boneh, ed., Lecture Notes in Comput. Sci. 2729, Springer-Verlag, Berlin, 2003, pp. 338–348, D. J. Bernstein, Math. Comp., 76 (2007), pp. 389–403].

Ideal semigroups of noetherian domains and Ponizovski decompositions

Let R be an integral domain with quotient field K and L ⊃ K a finite extension field. By an R -lattice in L we mean a finitely generated R -module containing a basis of L over K . The set of all R -lattices is a commutative multiplicative semigroup. If R is one-dimensional and noetherian, we determine the structure of this semigroup and of the corresponding class semigroup by means of its partial Ponizovski factors. If moreover R is a Dedekind domain and L ⊃ K is separable, we give criteria for the partial Ponizovski factors to be groups in terms of the different and the conductor of their endomorphism rings.