Elements of Number Theory

Elements of Number Theory


Note on Systems of Polynomial Equations Over Finite Fields

Abstract: "Let F be a finite field of q elements and characteristic p (so q = p[superscript n] for some n [> or =] 1) and let [gamma] := [formula] be a system of polynomial equations with coefficients in F. In this paper we relate the ...

Note on Systems of Polynomial Equations Over Finite Fields

Abstract: "Let F be a finite field of q elements and characteristic p (so q = p[superscript n] for some n [> or =] 1) and let [gamma] := [formula] be a system of polynomial equations with coefficients in F. In this paper we relate the structure of the F-algebra [formula] to the roots of [gamma] in F[superscript r]."

Certain Diagonal Equations Over Finite Fields

ABSTRACT: Let F q to the t be the finite field with q to the t elements and let F q to the t star be its multiplicative group.

Certain Diagonal Equations Over Finite Fields

ABSTRACT: Let F q to the t be the finite field with q to the t elements and let F q to the t star be its multiplicative group. We study the diagonal equation a times x to the (q minus 1) plus b times y to the (q minus 1) equals c, where a, b and c are elements of F q to the t star. This equation can be written as x to the (q minus 1) plus alpha times y to the (q minus 1) equals beta, where alpha and beta are elements of F q to the t star. Let N sub t (alpha, beta) denote the number of solutions (x, y) in F q to the t star cross F q to the t star of the equation x to the (q minus 1) plus alpha times y to the (q minus 1) equals beta and I(r;a, b) be the number of monic irreducible polynomials f with coefficients in F q of degree r with f(0) equals a and f(1) equals b. We show that N sub t (alpha, beta) can be expressed in terms of I(r;a, b), where r divides t and a, b are elements of F q star are related to alpha and beta. A recursive formula for I(r;a, b) will be given and we illustrate this by computing I(r;a, b) for r greater than or equal to 2 but less than or equal to 4. We also show that N sub 3 (alpha, beta) can be expressed in terms of the number of monic irreducible cubic polynomials over F q with prescribed trace and norm. Connsequently, N sub 3 (alpha, beta) can be expressed in terms of the number of rational points on a certain elliptic curve. We give a proof that given any a, b elements of F q star and integer r greater than or equal to 3, there always exists a monic irreducible polynomial f with coefficients in F q of degree r such that f(0) equals a and f(1) equals b. We also use the result on N sub 2 (alpha, beta) to construct a new family of planar functions.

Algorithmic Number Theory Efficient algorithms

7 Solving Equations over Finite Fields Solving Equations over Finite Fields In this chapter , we give efficient methods for solving polynomial equations over finite fields . Because the integers modulo a prime p form a field ...

Algorithmic Number Theory  Efficient algorithms

Volume 1.

Lacunary Polynomials Over Finite Fields

The text is a dependable reference for mathematicians and researchers interested in the study of reducible lacunary polynomials over finite fields.

Lacunary Polynomials Over Finite Fields

Lacunary Polynomials Over Finite Fields focuses on reducible lacunary polynomials over finite fields, as well as stem polynomials, differential equations, and gaussian sums. The monograph first tackles preliminaries and formulation of Problems I, II, and III, including some basic concepts and notations, invariants of polynomials, stem polynomials, fully reducible polynomials, and polynomials with a restricted range. The text then takes a look at Problem I and reduction of Problem II to Problem III. Topics include reduction of the marginal case of Problem II to that of Problem III, proposition on power series, proposition on polynomials, and preliminary remarks on polynomial and differential equations. The publication ponders on Problem III and applications. Topics include homogeneous elementary symmetric systems of equations in finite fields; divisibility maximum properties of the gaussian sums and related questions; common representative systems of a finite abelian group with respect to given subgroups; and difference quotient of functions in finite fields. The monograph also reviews certain families of linear mappings in finite fields, appendix on the degenerate solutions of Problem II, a lemma on the greatest common divisor of polynomials with common gap, and two group-theoretical propositions. The text is a dependable reference for mathematicians and researchers interested in the study of reducible lacunary polynomials over finite fields.

Encyclopedic Dictionary of Mathematics

For these miscellaneous results – Dickson [ 1 ] and Mordell [ 2 ] B. Equations over Finite Fields Let k be a finite field of characteristic p consisting of q ( = p ' ) elements . Chevalley's theorem : Let f be a tform of degree d in n ...

Encyclopedic Dictionary of Mathematics

V.1. A.N. v.2. O.Z. Apendices and indexes.

Finite Fields

Contemporary Mathematics Volume 168, 1994 New Results on Diagonal Equations over Finite Fields from Cyclic Codes Jacques Wolfmann ABSTRACT. Using a link between irreducible cyclic codes and certain diagonal equations over finite fields, ...

Finite Fields

Because of their applications in so many diverse areas, finite fields continue to play increasingly important roles in various branches of modern mathematics, including number theory, algebra, and algebraic geometry, as well as in computer science, information theory, statistics, and engineering. Computational and algorithmic aspects of finite field problems also continue to grow in importance. This volume contains the refereed proceedings of a conference entitled Finite Fields: Theory, Applications and Algorithms, held in August 1993 at the University of Nevada at Las Vegas. Among the topics treated are theoretical aspects of finite fields, coding theory, cryptology, combinatorial design theory, and algorithms related to finite fields. Also included is a list of open problems and conjectures. This volume is an excellent reference for applied and research mathematicians as well as specialists and graduate students in information theory, computer science, and electrical engineering.