Structure Theory of Fields

The Krull Galois theory for infinite separable normal extensions is generalized in this note to non-algebraic extensions. For any extension fieldEof a fieldKit is shown that the Galois groupGcan be given a translation invariant topology such that the closed subgroups are precisely the subgroups that figure in a Galois correspondence. For extension fieldsE/Ksuch thatE/Kis of finite transcendence degree and such thatEis Galois over each intermediate field the topology turns out to be compact and we have a Galois correspondence in the Krull fashion. For infinite transcendence degree extensions the Galois correspondence remains but compactness is lost. The topology coincides with the Krull topology in the case of algebraic extensions. Further properties of the topology are also studied.

Let K⊂M be a finite field extension. An intermediate field L is called “invariant” if there is an affine algebraic K-group acting on M with L as its invariant field. The question, which intermediate fields are invariant, was studied by Begueri [1] for purely inseparable extensions and by Sweedler [6] for arbitrary extensions, but only for a restricted class of groups. In this paper Begueri's result is generalized to arbitrary field extensions. Additionally it is shown that one can check whether a given intermediate field is invariant or not by computing the rank of certain matrices. As an application we get a class of invariant intermediate fields.

In this chapter, we investigate infinite Galois extensions and prove an analog of the fundamental theorem of Galois theory for infinite extensions. The key idea is to put a topology on the Galois group of an infinite dimensional Galois extension and then use this topology to determine which subgroups of the Galois group arise as Galois groups of intermediate extensions. We also give a number of constructions of infinite Galois extensions, constructions that arise in quadratic form theory, number theory, and Galois cohomology, among other places.

This paper proposes an exponentiation method with Frobenius mappings. The main target is an exponentiation in an extension field. This idea can be applied for scalar multiplication of a rational point of an elliptic curve defined over an extension field. The proposed method is closely related to so-called interleaving exponentiation. Unlike interleaving exponentiation methods, it can carry out several exponentiations of the same base at once. This happens in some pairing-based applications. The efficiency of using Frobenius mappings for exponentiation in an extension field was well demonstrated by Avanzi and Mihailescu. Their exponentiation method efficiently decreases the number of multiplications by inversely using many Frobenius mappings. Compared to their method, although the number of multiplications needed for the proposed method increases about 20%, the number of Frobenius mappings becomes small. The proposed method is efficient for cases in which Frobenius mapping cannot be carried out quickly.

In this chapter, we take a closer look at a finite field extension F < E from the point of view that E is a vector space over F. It is clear, for instance, that any σ ∈ G F(E) is a linear operator on E over F. However, there are many linear operators that are not field automorphisms. One of the most important is multiplication by a fixed element of E, which we study next.

Let K\k be a finite Galois extension of finite algebraic number fields with Galois group g. We denote by Gm the multiplicative group defined over the rational number field Q and put.

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

Symbolic computation of Drazin inverses by specializations

Let L be a number field and OL the ring of algebraic integers in L. For apolynomial f with coefficients in OL, the content of f in L is the ideal of OLgenerated by coefficients of f. The polynomial f is primitive in L if the contentof f in L is OL. In 2005, Arturo Magidin and David McKinnon proved the Gauss’ lemma fornumber fields, the product of two primitive polynomials is also primitive, andsome applications following from Gauss’ lemma for number fields. A function field K over a finite field k is a finite separable field extension overk(x) where x is a transcendantal element. In this research, we study Magidin and McKinnon’s work on the function fields.

Optimal normal bases are special cases of the so-called Gauss periods (Disquisitiones Arithmeticae, Articles 343---366); in particular, optimal normal bases are Gauss periods of type (n, 1) for any characteristic and of type (n, 2) for characteristic 2. We present the multiplication tables and complexities of Gauss periods of type (n, t) for all n and t = 3, 4, 5 over any finite field and give a slightly weaker result for Gauss periods of type (n, 6). In addition, we give some general results on the so-called cyclotomic numbers, which are intimately related to the structure of Gauss periods. We also present the general form of a normal basis obtained by the trace of any normal basis in a finite extension field. Then, as an application of the trace construction, we give upper bounds on the complexity of the trace of a Gauss period of type (n, 3).

We define a diagram associated with a purely inseparable field extension of finite exponent. We show that, under this definition, for any given field extension the shape of its diagram is unique. Thus our diagram improves the diagram defined by Sweedler in [2, p. 402]. In §2 we define an “allowable” shape for a diagram. Given any “allowable” shape for a diagram representing a finite field extension, we construct a field extension whose diagram has that shape.

Chudnovsky-type algorithms for the multiplication in finite extensions of finite fields are well-known for having a good bilinear complexity, both asymptotically and at finite distance. More precisely, for every degree n of the extension, the existence of a family of algorithms with linear bilinear complexity in n has been proved using the original method applied to an explicit recursive tower of function fields. However, there is currently no method to build these algorithms in polynomial time. Nevertheless, one can construct in polynomial time a Chudnovsky-type algorithm over the projective line for the multiplication in any extension degree, with a quasi-linear bilinear complexity. In this paper, we prove that we can obtain algorithms both constructible in polynomial time and having a linear bilinear complexity by mixing up these two strategies.

An isogeny class $\mathcal{A}$ of abelian varieties defined over finite fields is said to be &amp;quot;cyclic&amp;quot; if every variety in $\mathcal{A}$ has a cyclic group of rational points. In this paper we study the local cyclicity of Weil-central isogeny classes of abelian varieties, i.e. those with Weil polynomials of the form $f_\mathcal{A}(t)=t^{2g}+at^g+q^g$, as well as the local growth of the groups of rational points of the varieties in $\mathcal{A}$ after finite field extensions. We exploit the criterion: an isogeny class $\mathcal{A}$ with Weil polynomial $f$ is cyclic if and only if $f'(1)$ is coprime with $f(1)$ divided by its radical.

Direct computation of pth roots in extended finite fields of characteristic p ≥ 2 is introduced, wherein the reduction polynomial is irreducible and can be even random. Proposed method works in any case of p ≥ 2 and finite field extension. This method is the most efficient, it is even more efficient than the method, which is widely used, based on inversion of squaring matrix utilised in the case of p = 2. This method is more efficient regarding the computation and storing of the matrix as well as the computation of the roots.

Nilpotent and abelian Hopf–Galois structures on field extensions