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Dynamic Programming
Dynamic Programming

... •What is the base case? What is the recursive case? ...
Document
Document

Rings of constants of the form k[f]
Rings of constants of the form k[f]

3.1 15. Let S denote the set of all the infinite sequences
3.1 15. Let S denote the set of all the infinite sequences

Algebra 2 - TeacherWeb
Algebra 2 - TeacherWeb

05 Polynomials and Polynomial Functions
05 Polynomials and Polynomial Functions

Full text
Full text

... is that P n _! is then the greatest common divisor of a and b. The worst case, in the sense that the algorithm takes the longest possible number of iterations to terminate, is when the sequence a > r1 > r2 > ••• > vn = 0 decreases to 0 as slowly as possible. The smallest pairs (b9a) for which this h ...
Document
Document

... In the polynomial: x2y2 - 3x2y + 4xy2 - 2, there are no like terms. Even though three terms have xy’s in them, the powers are different. Remember the variables and the powers must match in order to be called like terms. ...
Equiangular Lines
Equiangular Lines

Write Numbers Using Prime Factorization
Write Numbers Using Prime Factorization

pi, fourier transform and ludolph van ceulen
pi, fourier transform and ludolph van ceulen

... Borweins' one must have efficient multiplication methods as well. It is the discrete Fast Fourier transform, which made fast multiplication of very long numbers possible. Almost all the current records apply one or another version of FFT multiplication. In recent years, the computation of the expans ...
Euclid`s Algorithm - Cleveland State University
Euclid`s Algorithm - Cleveland State University

... Euclidean algorithm has an upper bound on the number of steps it will take to find the GCD. This bound is found by the equation: k ≤ log2(a) + log2(b) where k is the number of steps. This inequality can be easily proven, if we assume a and b are positive integers and (a < b), we can replace b with r ...
Primes, Factorization, and the Euclidean Algorithm
Primes, Factorization, and the Euclidean Algorithm

Find the GCD of 2322 and 654
Find the GCD of 2322 and 654

... Least Common Multiple. First, you should know what a prime number is. It's a number that can't be expressed as the product (that means by multiplying together) smaller numbers. An example is 5. 5 can't be expressed as the product of any smaller numbers. Its only factors are 1 and 5. The Sieve of Era ...
Math 3390 Introduction to topology, Assignment 2. Due October 26
Math 3390 Introduction to topology, Assignment 2. Due October 26

Factors and Prime Factorization
Factors and Prime Factorization

x - Barnstable Academy
x - Barnstable Academy

Section 3 - Web4students
Section 3 - Web4students

Complex Numbers - Berkeley City College
Complex Numbers - Berkeley City College

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Full text

1.1 Polynomial Equations in Factored Form
1.1 Polynomial Equations in Factored Form

1/2 + square root 3/2i
1/2 + square root 3/2i

... Polynomial Functions • What are Polynomial functions? • Definition: Polynomial functions are defined and continuous on all real numbers. • Example of a polynomial function: f(x)= 4x^3-5x-1/2 (f is a polynomial function of degree 3 with leading coefficient ...
Dividing Polynomials
Dividing Polynomials

Algebraic Numbers and Algebraic Integers
Algebraic Numbers and Algebraic Integers

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PDF

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Factorization of polynomials over finite fields

In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to allow the computation of the factorization by means of an algorithm. In practice, algorithms have been designed only for polynomials with coefficients in a finite field, in the field of rationals or in a finitely generated field extension of one of them.The case of the factorization of univariate polynomials over a finite field, which is the subject of this article, is especially important, because all the algorithms (including the case of multivariate polynomials over the rational numbers), which are sufficiently efficient to be implemented, reduce the problem to this case (see Polynomial factorization). It is also interesting for various applications of finite fields, such as coding theory (cyclic redundancy codes and BCH codes), cryptography (public key cryptography by the means of elliptic curves), and computational number theory.As the reduction of the factorization of multivariate polynomials to that of univariate polynomials does not have any specificity in the case of coefficients in a finite field, only polynomials with one variable are considered in this article.
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