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Geometric Constructions from an Algebraic Perspective
Geometric Constructions from an Algebraic Perspective

... Theorem 24. Let α be a real number. Then α is constructible if and only if α belongs to the top of some square root tower over Q. Proof. (⇐)Let C be the set of constructible real numbers. C is an extension field of Q. C is a subfield of R because we have shown earlier that the constructible set C is ...
Graphing with Asymptotes
Graphing with Asymptotes

Exponential sums with multiplicative coefficients
Exponential sums with multiplicative coefficients

1 Prime numbers
1 Prime numbers

LINEAR INDEPENDENCE OF LOGARITHMS OF - IMJ-PRG
LINEAR INDEPENDENCE OF LOGARITHMS OF - IMJ-PRG

... a1 · · · abmm − 1 ≥ A−nB where A = max{2, a1 , . . . , an } and B = max{2, |b1 |, . . . , |bn |}; this is a kind of Liouville estimate: the absolute value of a non-zero rational number is at least the inverse of a denominator. This estimate is sharp in terms of A and n, but not in terms of B; ther ...
19 Feb 2010
19 Feb 2010

Revision 2 - Electronic Colloquium on Computational Complexity
Revision 2 - Electronic Colloquium on Computational Complexity

Elements of Modern Algebra
Elements of Modern Algebra

... A minimal amount of mathematical maturity is assumed in the text; a major goal is to develop mathematical maturity. The material is presented in a theorem-proof format, with definitions and major results easily located thanks to a user-friendly format. The treatment is rigorous and self-contained, i ...
Algebra: Monomials and Polynomials
Algebra: Monomials and Polynomials

+ 1 - Stefan Dziembowski
+ 1 - Stefan Dziembowski

Strategies for Proofs
Strategies for Proofs

PDF of Version 2.01-B of GIAA here.
PDF of Version 2.01-B of GIAA here.

Light leaves and Lusztig`s conjecture 1 Introduction
Light leaves and Lusztig`s conjecture 1 Introduction

DM- 07 MA-217 Discrete Mathematics /Jan
DM- 07 MA-217 Discrete Mathematics /Jan

The maximum modulus of a trigonometric trinomial
The maximum modulus of a trigonometric trinomial

elementary number theory - School of Mathematical Sciences
elementary number theory - School of Mathematical Sciences

a) - BrainMass
a) - BrainMass

... a) Define the greatest common divisor of two integers. Let a and b be two integers . Then the greatest common divisor of a and b is defined as the largest positive integer k such that a and b are both divisible by k. Denoted by GCD(a,b). For instance, the gcd(2,4)=2 ...
A First Course in Abstract Algebra: Rings, Groups, and Fields
A First Course in Abstract Algebra: Rings, Groups, and Fields

Unit 2 Scholar Study Guide Heriott-Watt
Unit 2 Scholar Study Guide Heriott-Watt

The ordered distribution of Natural Numbers on the Square Root Spiral
The ordered distribution of Natural Numbers on the Square Root Spiral

Intro Abstract Algebra
Intro Abstract Algebra

Intro Abstract Algebra
Intro Abstract Algebra

Full text
Full text

... for a natural algebraic and geometric setting for their analysis. In this way many known results are unified and simplified and new results are obtained. Some of the results extend to Fibonacci representations of higher order, but we do not present these because we have been unable to extend the the ...
ON SEQUENCES DEFINED BY LINEAR RECURRENCE
ON SEQUENCES DEFINED BY LINEAR RECURRENCE

4 Number Theory 1 4.1 Divisors
4 Number Theory 1 4.1 Divisors

... Note: This definition satisfies gcd(0,1) = 1. The lowest common multiple lcm(a,b) is defined as follows: lcm(a,b) = min(m > 0 s.t. a|m and b|m) (for a 6= 0 and b 6= 0). a and b are coprimes (or relatively prime) iff gcd(a,b) = 1. Prime Numbers An integer p ≥ 2 is called prime if it is divisible only ...
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Factorization



In mathematics, factorization (also factorisation in some forms of British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomial x2 − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained.The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viète's formulas relate the coefficients of a polynomial to its roots.The opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms.Integer factorization for large integers appears to be a difficult problem. There is no known method to carry it out quickly. Its complexity is the basis of the assumed security of some public key cryptography algorithms, such as RSA.A matrix can also be factorized into a product of matrices of special types, for an application in which that form is convenient. One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types: QR decomposition, LQ, QL, RQ, RZ.Another example is the factorization of a function as the composition of other functions having certain properties; for example, every function can be viewed as the composition of a surjective function with an injective function. This situation is generalized by factorization systems.
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