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Chap 6
Chap 6

... In other words, no matter which elements in each sector we choose, the sums and products are equivalent. This allows us to compute sums and products by choosing any representatives we want. Example 6.2. Referring to the equivalence classes for m = 5 of Example 6.1 and the letters of Theorem 5.3, tak ...
STRUCTURE THEOREMS OVER POLYNOMIAL RINGS 1
STRUCTURE THEOREMS OVER POLYNOMIAL RINGS 1

... Condition (1) of the theorem is independent of the ring R, so if, for given S, k and G, one of the other conditions is satisfied for some ring R then it is also satisfied by any other ring R satisfying the hypotheses of the theorem. We thank the referee for the elegant proofs of 3.3 and 8.1. 2. Chan ...
Homomorphisms, ideals and quotient rings
Homomorphisms, ideals and quotient rings

Full text
Full text

... testing if a number is representable as a sum of three cubes, nor a method for finding such a representation (see [12]). For the quadratic forms, the situation is different; Gauss developed his theory of quadratic forms [7] and solved the related integer representation problem. In particular, quadra ...
DEGREE OF REGULARITY FOR HFE
DEGREE OF REGULARITY FOR HFE

... consisting of polynomials p1 (x1 , . . . , xn ), . . . , pn (x1 , . . . , xn ) is the lowest degree at which we have non-trivial polynomial relations between the pi components. It is commonly accepted that in general this is the degree at which the solving algorithm will terminate and therefore it i ...
A quantitative lower bound for the greatest prime factor of (ab + 1)(bc
A quantitative lower bound for the greatest prime factor of (ab + 1)(bc

Basic Algorithmic Number Theory
Basic Algorithmic Number Theory

... We give general definitions for the success probability of an algorithm in this section, but rarely use the formalism in our later discussion. Instead, for most of the book, we focus on the case of algorithms that always succeed (or, at least, that succeed with probability extremely close to 1). Thi ...
Homework #5 Solutions (due 10/10/06)
Homework #5 Solutions (due 10/10/06)

... Yet another way of expressing this is that NG may be regarded as a function on the set of conjugacy classes of subgroups. Now we note that almost all of our subgroups can be identified as either cyclic subgroups or as certain normalizers (or centralizers). Cyclic subgroups are easily divided into c ...
Part XV Appendix to IO54
Part XV Appendix to IO54

... (2) The group (G , ◦) is called cyclic if it contains an element g , called generator, such that the order of (g ) = |G |. Theorem If the multiplicative group (Zn? , ×n ) is cyclic, then it is isomorphic to the additive group (ZΦ(n) , +Φ(n) ). (However, no effective way is known, given n, to create ...
the arithmetical theory of linear recurring series
the arithmetical theory of linear recurring series

... is self-explanatory. The following convenient definition was introduced by H. T. Engstrom*: A number ir is said to be a general period of the difference equation (1.1) for the modulus m if every sequence of rational integers (m) satisfying (1.1) has the period ir. Let t be the least such general per ...
Quotient Rings of Noncommutative Rings in the First Half of the 20th
Quotient Rings of Noncommutative Rings in the First Half of the 20th

(pdf)
(pdf)

(pdf)
(pdf)

... Example 2.20. Recalling the algebraic integer 2 + 3 = α from example 2.8, we compute NQ (α). This amounts to taking the determinant of the linear transformation in 2.8, or, because we are lazy, simply plugging in 0 for the minimal polynomial of α in the determinant already computed. Then N (α) = 1, ...
Section 0. Background Material in Algebra, Number Theory and
Section 0. Background Material in Algebra, Number Theory and

... numbers are in the same coset iff they have the same modulus. Clearly, the left cosets are just the circles with centre 0, and these are the elements of G/H. We have ‘modded out’ by H, removing the argument information, and retaining only the modulus information. We can turn G/H into a group under m ...
Chapter V. Solvability by Radicals
Chapter V. Solvability by Radicals

roots of unity - Stanford University
roots of unity - Stanford University

Improving the Effectiveness of Marketing and Sales using Genetic
Improving the Effectiveness of Marketing and Sales using Genetic

2.1, 2.3-2.5 Review
2.1, 2.3-2.5 Review

... An asset with a first cost of $100,000 is depreciated over 5-year period. It is expected to have a $10,000 salvage value at the end of 5 years. Using the straight-line method, what is the book value at the end of year 2? ...
Precalculus PreAP/D Rev 2017 2.5: Rational Zero Test “I WILL
Precalculus PreAP/D Rev 2017 2.5: Rational Zero Test “I WILL

EIGENVALUES OF PARTIALLY PRESCRIBED
EIGENVALUES OF PARTIALLY PRESCRIBED

... This paper is a natural generalization of those results. As the main result (Theorem 3.1), we give a complete solution of Problem 1.1 in the case when the eigenvalues of the matrix (1.2) belong to F, and F is an infinite field. In particular, this gives the complete solution of Problem 1.1 over algebr ...
p-adic Heights of Heegner Points and Anticyclotomic
p-adic Heights of Heegner Points and Anticyclotomic

Primality - Factorization
Primality - Factorization

Ring Theory
Ring Theory

Number Fields
Number Fields

A potential relation between the algebraic approach to calculus and
A potential relation between the algebraic approach to calculus and

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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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