Algebraic Field Extension Research Articles

Polynomials Solving Dirichlet Boundary Value Problems

Let p be an odd prime. Then -1 is a quadratic residue (mod p) if and only if p 1 (mod 4), and 3 is a quadratic residue (mod p) if and only if p = 1 (mod 6). These results are well known. The first follows immediately from Euler's criterion and it appears as a theorem in just about every book in which quadratic residues are discussed. The proof of the result about -3 usually requires a little more effort and the law of quadratic reciprocity. The purpose of this note is to show that the if' part of both results follows from Lagrange's theorem in group theory and the theory of algebraic field extensions. Let Zp be the field of integers (mod p). Assume that -1 is not a square (quadratic residue) in Zp. The quadratic extension Zp( -1) has p2 elements. Thus its group of nonzero elements Zp( -)* has p2_ 1 elements and the subgroup Zp* (the group of nonzero elements in Zp) has p 1 elements. The group we want to examine is the factor group Zp( -1)*/Zp*, with p + 1 elements. A direct calculation shows that the coset represented by 1 + has order 4. Thus 4 divides p + 1, which means that p -1 (mod 4). To prove the result about -3, extend Zp to Zp( -3). The order of the coset represented by 3 + -3 in the factor group Zp(-30)*/Zp* is 6. So, again by Lagrange's theorem, p -1 (mod 6).

Some Relationship between Anti-Integral Extensions of Noetherian Domains

Let R be a Noetherian domain with quotient field K and let α be an anti-integral element of degree d over R. Let β be an elemen of R(α) (resp. R(α,α-1)) such that β is an anti-integral element over R and that R(α) (resp. R(α,α-1)) is integral over R(β)). We shall investigate some properties descending from R(α) (resp. R(α,α-1)) to R(β), i. e., flatness and faithful flatness, and study the ideals J(α), J(β), J(α) and J(β). Let R be a Noetherian domain and R(X) a polynomial ring. Let α be an element of an algebraic extension field L of the quotient field K of R and let π: R(X) →R(α) be the R-algebra homomorphism, sending X to α. Let ψα(X) be the monic minimal polynomial of α over K with deg ψα(X)=d and write

One-Way Communication Complexity of Computing a Collection of Rational Functions

We consider the problem of evaluating a collection of rational functions ƒ 1( x, y), ƒ 2( x, y), . , ƒ s ( x, y) (x ∈ R m , y ∈ R n ) using two processors P 1 and P 2, assuming that processor P 1 (respectively P 2) has access to input x (respectively y) and the functional form of ƒ. We establish, by way of algebraic field extension theory, an almost optimal lower bound on the one-way communication complexity (i.e.. the minimum number of real-valued messages that have to be exchanged). Our result strengthens the early result of Abelson in several directions.

An embedding of the Poincaré Lie algebra into an extension of the Lie field of SO0(1,4)

The * isomorphism between algebraic extensions of the Lie fields of SO0(1,4) and the Poincaré group are considered herein. It is shown that the principal series of unitary ray representations of SO0(1,4) are associated, via the * isomorphism, with real mass, positive and negative energy representations of the Poincaré Lie algebra with arbitrary spin. Results on the most degenerate exceptional series of SO0(1,4) representations are also given.

On the communication complexity of distributed algebraic computation

Abstract : we consider a situation where two processors P sub 1 and P sub 2 ar e to evaluate a collection of functions function 1,....,function 8 of two vector variables x, y, under the assumption that processor P sub 1 (respectively, P sub 2) has access only to the value of the variable x (respectively, y) and the functional form of function 1,....,function 8. We provide some new bounds on the communication complexity (the amount of information that has to be exchanged between the processors) for this problem. An almost optimal bound is derived for the case of one-way communication when the functions function 1,...., function 8 are polynomials. We also derive some new lower bounds for the case of two-way communication which improve on earlier bounds by Abelson A 80. As an application, we consider the case where x and y are n x n matrices and f(x,y) is a particular entry of the inverse of x + y. Under certain restriction on the class of allowed communication protocols, we obtain an omega(n squared) lower bound, in contrast to the omega(n) lower bound obtained by applying Abelson's results. Our results are based on certain tools from classical algebraic geometry and field extension theory.

On simple periodic linear groups— Dense subgroups, permutation representations, and induced modules

We determine the Zariski-dense subgroups of Chevalley groups and their twisted analogues over infinite algebraic extensions of finite fields. It turns out that these are essentially forms of the same group (possibly becoming twisted) over smaller infinite fields. It follows from our classification that if\(\bar G\) is a simple algebraic group over the algebraic closure of a finite field, then a dense subgroup of\(\bar G\) can never be maximal, and so the maximal subgroups of\(\bar G\) are necessarily closed. It follows that Seitz’s determination of the closed maximal subgroups of\(\bar G\) actually gives all the maximal subgroups.

High degree anti-integral extensions of Noetherian domains

Introduction. Let R be a Noetherian integral domain and R [X] a polynomial ring. Let a be an element of an algebraic field extension L of the quotient field K of R and let π : R [X] -> R [a] be the Λ-algebra homomorphism sending X to a. Let φΛ(X) be the monic minimal polynomial of a over K with deg φΛ(X)=d and write φΛ(X)=X d+ηlX -+ +ηd. Let 7ω:= Π ί,ι(R:R ?,). Foτf(X)^R[X], let C(f(X)) denote the ideal generated by the coefficients of f ( X ) . Let/[Λ]: =/[*] C(φΛ(X)), which is an ideal of R and contains /[*]. The element a is called an anti-integral element of degree d over R if Kerτr= /[rt] φΛ(^C) R [X]. When a is an anti-integral element over Ry R[a] is called an anti-integral extension of R. In the case K(a)=K, an anti-integral elemet a is the same as an anti-integral element (i.e., R=R[a] Γ\R[l/(X\) defied in [5]. The element a is called a super-primitive element of degree d over R if JιΛ^p for all primes p of depth one. For p^ Spec (R), k(p) denotes the residue field Rp/pRp and rank^) R [a] ®R k(p) denotes the dimension as a vector space over k(p). We are interested in characterizing the flatness and the integrality of an anti-integral extension R\cί of R. Indeed, among others we obtain the following results: (i) R [a] is flat over R if and only if rank^) R [a] ®R k(p)<d for all p <Ξ Sρec(.R), (ii) R[a] is integral over R if and only if ra.nik(p)R[a](&R k(p)=d for all

Fuzzy algebraic field extensions

In this paper we introduce the concept of fuzzy algebraic field extensions. We determine conditions for which a fuzzy algebraic field extension has unique maximal fuzzy intermediate fields which are fuzzy separable algebraic and fuzzy purely inseparable.

Algebraic number field extensions with prescribed trace form

Algebraic number field extensions with prescribed trace form

Rings of formal power series with homeomorphic prime spectra

LetR⊂T be domains, not fields, such that Spec(R)=Spec(T) as sets; that is, such that the prime ideals ofT coincide, as sets, with those ofR. It is proved that the canonical map Spec(T[[X]])→Spec(R[[X]]) is a homeomorphism. This generalizes a result of Girolami in caseR is a pseudovaluation domain with the SFT (strong finite type)—property andT is its associated valuation domain. The analogous property for polynomial rings is also characterized: Spec(T[X])→Spec(R[X]) is a homeomorphism if and only ifR/M⊂T/M is a purely inseparable (algebraic) field extension, whereM is the maximal ideal ofR.

Construction of normal bases of rings of integral elements of certain fields of algebraic functions

This article is devoted to the construction of normal bases of rings of integral elements of certain fields of algebraic functions. These fields are algebraic extensions of the initial fields, given by appropriate algebraic equations. The extensions are given only by the monodromy group. The device of the matrix Riemann boundary-value problem on a Riemann surface is used.

Computing primitive elements of extension fields

Several mathematical results and new computational methods are presented for primitive elements and their minimal polynomials of algebraic extension fields. For a field Q( α 1 ,…, α t ) obtained by adjoining algebraic numbers α 1 ,… α t to the rational number field Q, it is shown that there exists at least one vector =( s 1 ,…, s t ) of integers in a specially selected set of (−1) N vectors such that s 1 α 1 + s 2 α 2 +…+ s t α t is a primitive element, where N is the degree of Q( α 1 ,…, α t ) over Q. Furthermore, a method is presented for directly calculating such a vector, that gives a primitive element. Finally, for a given polynomial f over Q, a new method is presented for computing a primitive element of the splitting field of f and its minimal polynomial over Q.

On extensions of higher derivations for algebraic extensions of fields of positive characteristics

On extensions of higher derivations for algebraic extensions of fields of positive characteristics

On a Number-Theoretic Result of Sylvester-Kronecker-Zsigmondy

Let K be an algebraic extension field of Q. Further, let 𝔒K be the domain of algebraic integers of K, and let Σn(x) be the n-th cyclotomic polynomial. This paper is devoted to the factorization of the principal ideal Σn(e) 𝔒K(e𝔒K) into prime ideals of 𝔒K. The main result (Theorem 3.4) can be considered as a generalization of a known result of Sylvester, Kronecker and Zsigmondy on the prime factorization of Σn(e) (eϵZ). With Theorem 3.4., we improve corresponding results of Redei [11] and Sachs [14]. We generalize a technique developed in [8] and [3] and we study also the cases that K is a quadratic field and a cyclotomic field, respectively. Finally, we apply the results to the parameter determination of Fourier-like number-theoretic transforms in 𝔒K.

On the construction of independent units in cyclic or radical extensions

A general theorem on independency in a ℤ[G]-module is proved where G is a finite cyclic group. If applied to radical algebraic extensions of algebraic number fields, it yields a criterion of independency for systems of (parametric) units which contains as special cases theorems by Frei and Levesque and Halter-Koch and Stender.

Local differents of algebraic and finite extensions of valued fields

The properties of discriminants and differents were studied first by Dedekind and Hilbert in finite algebraic extensions of fields of algebraic numbers. From a local point of view, that is equivalent to a study of the p-adic case, where the results of Dedekind and Hilbert can be formulated as follows. Dedekind's theorem: The g.c.d. Δ( K k ) of differents of integral bases of a finite algebraic extension K k (which I call an algebraic different if K k and the g.c.d δ( K k ) of differents of integral elements of K k (which I call an arithmetic different of K k ) coincide; Hilbert's theorem (which is the basis of Herbrand's ramification theory of intermediate extensions): If K ⊃ L ⊃ k, δ( K k ) = δ( K L ) δ( L k ) . These results are easily generalizable to the “classical case” of henselian valued basic fields, i.e., the case when the valuation is discrete and the residual extension K k of K k is separable. But, in the general case of extensions K k of valued fields (where k may be assumed to be henselian), Dedekind's and Hilbert's theorems are not always true: the algebraic different Δ( K k ) divides the arithmetic different δ( K k ) , but generally δ( K k ) ≠ Δ( K k ) , and Hilbert's theorem holds only for the algebraic different. When the valuation is discrete, I call an extension K k dedekindian when δ( K k ) = Δ( K k ) and hilbertian if, for every intermediate field L of K k (i.e., K ⊇ L ⊇ k), Hilbert's theorem δ( K k ) = δ( K L ) δ( L k ) for arithmetic differents holds. When the valuation is dense, the situation is more complicated, because of the existence of two kinds of ideals (principal and other), and it is convenient to define dedekindian and hilbertian extensions in a slightly different manner and to introduce somewhat wider classes of extensions called quasi-dedekindian and quasi-hilbertian. I study the relations between Δ( K k ) and δ( K k ) , and, in particular, I give a complete characterization of dedekindian extensions for both discrete and dense valuations; I also give examples of non-dedekindian and non-hilbertian extensions. In Section 4, some connections with the ramification theory (both for normal and non-normal extensions) are studied and a weak analog of Hilbert's theorem [ δ( K k ) δ( L k ) divides δ( K k ) ] is proved.

Universally Incomparable Ring-Homomorphisms

A homomorphism f: R → T of (commutative) rings is said to be universally incomparable in case each base change R → S induces an incomparable map S → S⊗RT. The most natural examples of universally incomparable homomorphisms are the integral maps and radiciel maps. It is proved that a homomorphism f: R → T is universally incomparable if and only if f is an incomparable map which induces algebraic field extensions of fibres, k(f-1(Q))→k(Q), for each prime ideal Q of T. In several cases (f algebra-finite, T generated as R-algebra by primitive elements, T an overring of a one-dimensional Noetherian domain R), each universally incomparable map is shown to factor as a composite of an integral map and a special kind of radiciel.

Ordered fields with the dense orbits property

For a formally real field K let X(K) be the space of orderings. The automorphism group Aut(K) of K operates on X(K). K is said to have the ’dense orbits property’ (DOP) if for all 2 X(K) the orbit of is dense in X(K). The study of such fields, which was begun in D. W. Dubois and the second author [Contemp. Math. 8, 265-288 (1982; Zbl 0484.12003)], is continued here. The main question considered is the behavior of the DOP under purely transcendental and under algebraic field extensions.

Certain systems of bilinear forms whose minimal algorithms are all quadratic

In this paper we prove that every minimal algorithm for computing a bilinear form is quadratic. It is then easy to show that for certain systems of bilinear forms all minimal algorithms are quadratic. One such system is for computing the product of two arbitrary elements in a finite algebraic extension field. This result, together with the results of the author in (E. Feig, to appear) and those of Winograd ( Theoret. Comput. Sci. 8 (1979), 359–377), completely characterize all minimal algorithms for computing products in these fields.

On systems of bilinear forms whose minimal division-free algorithms are all bilinear

In this paper we prove that for a certain class of systems of bilinear forms, all minimal division-free algorithms are essentially bilinear. This class includes systems for computing products in finite algebraic extension fields, and systems for computing the products of Toeplitz and Hankel matrices with vectors. Our results, together with the classification theorems of S. Winograd ( Theoret. Comput. Sci. 8 (1979), 359–377; Math. Systems Theory 10 (1977) , 169–180) completely describe all minimal division-free algorithms for computing these systems. We also prove, as an immediate consequence of our results, that the multiplicative complexity of the quaternion product over a real field is 8.