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Abstract Algebra - UCLA Department of Mathematics
Abstract Algebra - UCLA Department of Mathematics

solutions to the first homework
solutions to the first homework

Section2.2notes
Section2.2notes

Modern Algebra: An Introduction, Sixth Edition
Modern Algebra: An Introduction, Sixth Edition

The Theory of Polynomial Functors
The Theory of Polynomial Functors

A First Course in Abstract Algebra: Rings, Groups, and Fields
A First Course in Abstract Algebra: Rings, Groups, and Fields

Elements of Modern Algebra
Elements of Modern Algebra

... that could be used to solve problems of the same type, and treatments were generalized to deal with whole classes of problems rather than individual ones. In our study of abstract algebra, we shall make use of our knowledge of the various number systems. At the same time, in many cases we wish to ex ...
Notes on Galois Theory
Notes on Galois Theory

... Definition 1.1. Let E be a field. An automorphism of E is a (ring) isomorphism from E to itself. The set of all automorphisms of E forms a group under function composition, which we denote by Aut E. Let E be a finite extension of a field F . Define the Galois group Gal(E/F ) to be the subset of Aut ...
A Readable Introduction to Real Mathematics
A Readable Introduction to Real Mathematics

Numbers, Groups and Cryptography Gordan Savin
Numbers, Groups and Cryptography Gordan Savin

Factorising - Numeracy Workshop
Factorising - Numeracy Workshop

... Email: [email protected] [email protected] ...
Information Protection Based on Extraction of Square Roots of
Information Protection Based on Extraction of Square Roots of

... Hence, on the decryption stage, if Bob gets (2755, 3466) and (3466, 2755), there is no way to decide which of two Gaussians is authentic. In general, for p = 6221 there are several hundreds Gaussians that have this property, which creates ambiguity. Indeed, consider a Gaussian with three-digit compo ...
Lecture Notes for Math 614, Fall, 2015
Lecture Notes for Math 614, Fall, 2015

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

CParrish - Mathematics
CParrish - Mathematics

... (GCD) of a and b. We will denote a GCD of a and b by gcd(a, b). (You may be somewhat mystified by what seems to be an effort to make a simple concept appear more complex. Please be patient; you will see that this definition of a GCD will generalize readily to other mathematical entities, which in so ...
PDF Polynomial rings and their automorphisms
PDF Polynomial rings and their automorphisms

Results on Some Generalizations of Interval Graphs
Results on Some Generalizations of Interval Graphs

... v ∈ V (G), v is not in one of N [X], N [Y ], or N [Z]. Thus N [X] ∩ N [Y ] ∩ N [Z] = ∅. ⇐ Suppose G is an n − star graph with middle clique C. Let (P, <) be the middle clique partial order for middle clique C. Suppose (P, <) has a covering with three chains with top elements X, Y, Z such that N [X] ...
Lecture 5 Message Authentication and Hash Functions
Lecture 5 Message Authentication and Hash Functions

... • if x is composite it outputs yes with probability at most ¼. Probability is taken only over the internal randomness of the algorithm, so we can iterate! The error goes to zero exponentially fast. This algorithm is fast and practical! ...
Algebraic Shift Register Sequences
Algebraic Shift Register Sequences

, Elementary Number Theory
, Elementary Number Theory

... where a,b, and m are large numbers All computations will involve repeated modular multiplications (one multiplication followed by one division by the modulus m, to get the remainder) If modulus m is a n-bit number the largest possible value after a multiplication is guaranteed to be less than 2nbits ...
Algebraic Number Theory, a Computational Approach
Algebraic Number Theory, a Computational Approach

Number Theory - FIU Faculty Websites
Number Theory - FIU Faculty Websites

Algebraic Number Theory, a Computational Approach
Algebraic Number Theory, a Computational Approach

... groups, and such a presentation can be reinterpreted in terms of matrices over the integers. Next we describe how to use row and column operations over the integers to show that every matrix over the integers is equivalent to one in a canonical diagonal form, called the Smith normal form. We obtain ...
local version - University of Arizona Math
local version - University of Arizona Math

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

... codes and PCPs of proximity). We define a concrete-efficiency threshold that indicates the smallest problem size beyond which the PCP becomes “useful”, in the sense that using it actually pays off relative to naive verification by simply rerunning the computation; our definition takes into account b ...
1 2 3 4 5 ... 46 >

Polynomial greatest common divisor

In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common divisor of two integers.In the important case of univariate polynomials over a field the polynomial GCD may be computed, like for the integer GCD, by Euclid's algorithm using long division. The polynomial GCD is defined only up to the multiplication by an invertible constant.The similarity between the integer GCD and the polynomial GCD allows us to extend to univariate polynomials all the properties that may be deduced from Euclid's algorithm and Euclidean division. Moreover, the polynomial GCD has specific properties that make it a fundamental notion in various areas of algebra. Typically, the roots of the GCD of two polynomials are the common roots of the two polynomials, and this allows to get information on the roots without computing them. For example, the multiple roots of a polynomial are the roots of the GCD of the polynomial and its derivative, and further GCD computations allow to compute the square-free factorization of the polynomial, which provides polynomials whose roots are the roots of a given multiplicity.The greatest common divisor may be defined and exists, more generally, for multivariate polynomials over a field or the ring of integers, and also over a unique factorization domain. There exist algorithms to compute them as soon as one has a GCD algorithm in the ring of coefficients. These algorithms proceed by a recursion on the number of variables to reduce the problem to a variant of Euclid's algorithm. They are a fundamental tool in computer algebra, because computer algebra systems use them systematically to simplify fractions. Conversely, most of the modern theory of polynomial GCD has been developed to satisfy the need of efficiency of computer algebra systems.
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