Hovhannes Khudaverdian's notes
... Definition. We say that a field K has a characteristic 0 if its prime subfield is Q. We say that a field K has a characteristic p if its prime subfield is Fp . Geometric interpretaion of the ring Z/N Z. This ring can be interpreted in terms of regular N -gon on the plane. Namely consider n {A0 , A1 ...
... Definition. We say that a field K has a characteristic 0 if its prime subfield is Q. We say that a field K has a characteristic p if its prime subfield is Fp . Geometric interpretaion of the ring Z/N Z. This ring can be interpreted in terms of regular N -gon on the plane. Namely consider n {A0 , A1 ...
RSK Insertion for Set Partitions and Diagram Algebras
... The irreducible representations of Sn are indexed by integer partitions of n. If λ = (λ1 , . . . , λ` ) ∈ (Z≥0 )` with λ1 ≥ · · · ≥ λ` and λ1 + · · · + λ` = n, then λ is a partition of n, denoted λ ` n. If λ ` n, then we write |λ| = n. If λ = (λ1 , · · · , λ` ) and µ = (µ1 , . . . , µ` ) are partiti ...
... The irreducible representations of Sn are indexed by integer partitions of n. If λ = (λ1 , . . . , λ` ) ∈ (Z≥0 )` with λ1 ≥ · · · ≥ λ` and λ1 + · · · + λ` = n, then λ is a partition of n, denoted λ ` n. If λ ` n, then we write |λ| = n. If λ = (λ1 , · · · , λ` ) and µ = (µ1 , . . . , µ` ) are partiti ...
Factoring Polynomials
... This lesson provides an opportunity to address Mathematical Practice MP.8, which calls for students to “look for and express regularity in repeated reasoning.” Students are already familiar with multiplying polynomials but, in this lesson, they must analyze the conditions that help them factor a pol ...
... This lesson provides an opportunity to address Mathematical Practice MP.8, which calls for students to “look for and express regularity in repeated reasoning.” Students are already familiar with multiplying polynomials but, in this lesson, they must analyze the conditions that help them factor a pol ...
The University of Sydney MATH 1004 Second Semester Discrete
... These two things that we are taking as obvious rest on laws of logic, which will be discussed later (in Chapter 11 of Choo and Taylor). Proof, Part 1. Suppose that x ∈ X \ (A ∩ B). Then x ∈ X and x ∈ / A ∩ B. Since x ∈ / A ∩ B, x ∈ / A or x ∈ / B. Suppose that x ∈ / A. Since we know x ∈ X, then x ∈ ...
... These two things that we are taking as obvious rest on laws of logic, which will be discussed later (in Chapter 11 of Choo and Taylor). Proof, Part 1. Suppose that x ∈ X \ (A ∩ B). Then x ∈ X and x ∈ / A ∩ B. Since x ∈ / A ∩ B, x ∈ / A or x ∈ / B. Suppose that x ∈ / A. Since we know x ∈ X, then x ∈ ...
Polynomials
... to the kind of polynomials one is generally working with than to individual polynomials; for instance when working with univariate polynomials one does not exclude constant ...
... to the kind of polynomials one is generally working with than to individual polynomials; for instance when working with univariate polynomials one does not exclude constant ...
Checking Polynomial Identities over any Field: Towards a
... dure used frequently in this context. In addition, many results in learning theory, and sparse multivariate polynomial interpolation also rely on checking polynomial identities [Zip79, GKS90, CDGK91, RB91]. Clearly, the problem is easy if the input polynomials are given as lists of coefficients (kno ...
... dure used frequently in this context. In addition, many results in learning theory, and sparse multivariate polynomial interpolation also rely on checking polynomial identities [Zip79, GKS90, CDGK91, RB91]. Clearly, the problem is easy if the input polynomials are given as lists of coefficients (kno ...
GENERALIZED CAYLEY`S Ω-PROCESS 1. Introduction We assume
... The correct category of representations of an affine algebraic monoid M , is the so called category of polynomial M -modules. If V is a finite dimensional k–space, we say that an action M × V → V is polynomial if the associated map M → Endk (V ) is a morphism of affine algebraic monoids, in particul ...
... The correct category of representations of an affine algebraic monoid M , is the so called category of polynomial M -modules. If V is a finite dimensional k–space, we say that an action M × V → V is polynomial if the associated map M → Endk (V ) is a morphism of affine algebraic monoids, in particul ...
Polynomials for MATH136 Part A
... God, it has been said, gave us the natural numbers 1,2,3,... and man created the rest. When we just had the natural numbers 1,2,3,... we had to say that the equation 2x = 1 has no solutions. So fractions were invented. Then 2x = 1 could be solved but x 2 = 2 could not. So along come the irrational n ...
... God, it has been said, gave us the natural numbers 1,2,3,... and man created the rest. When we just had the natural numbers 1,2,3,... we had to say that the equation 2x = 1 has no solutions. So fractions were invented. Then 2x = 1 could be solved but x 2 = 2 could not. So along come the irrational n ...
Algebra IIA Unit III: Polynomial Functions Lesson 1
... The degree of a monomial is found by _______________________________________________________ A leading coefficient is ___________________________________________________________________ A polynomial function is __________________________________________________________________ ...
... The degree of a monomial is found by _______________________________________________________ A leading coefficient is ___________________________________________________________________ A polynomial function is __________________________________________________________________ ...
A.2 Polynomial Algebra over Fields
... our alphabet. Additive structure has been provided by considering words as members of the vector space F n of n-tuples from F for appropriate n. Scalar multiplication in F n does not however provide a comprehensive multiplication for words and vectors. In order to construct a workable definition of ...
... our alphabet. Additive structure has been provided by considering words as members of the vector space F n of n-tuples from F for appropriate n. Scalar multiplication in F n does not however provide a comprehensive multiplication for words and vectors. In order to construct a workable definition of ...
Notes and Exercises
... i. In how many ways can a committee of 3 ladies and 2 men be selected from a group of 7 ladies and 5 men? ii. A committee of 1 girl and 2 boys is to be formed out of 5 girls and 6 boys. In how many ways can this be done? iii. In how many ways can two red balls and three black balls be chosen from a ...
... i. In how many ways can a committee of 3 ladies and 2 men be selected from a group of 7 ladies and 5 men? ii. A committee of 1 girl and 2 boys is to be formed out of 5 girls and 6 boys. In how many ways can this be done? iii. In how many ways can two red balls and three black balls be chosen from a ...
Polynomials
... The degree of a polynomial is the highest x power in the expression. Add or subtract polynomials by column addition or subtraction, or by collecting like terms. Multiply polynomials using any method that helps you to remember to multiply every term in one expression by every term in the other. Solve ...
... The degree of a polynomial is the highest x power in the expression. Add or subtract polynomials by column addition or subtraction, or by collecting like terms. Multiply polynomials using any method that helps you to remember to multiply every term in one expression by every term in the other. Solve ...
Polynomials and Taylor`s Approximations
... In elementary algebra, quadratic formula are given for solving all second degree polynomial equations in one variable. There are also formulae for the cubic and quartic equations. For higher degrees, Abel–Ruffini theorem asserts that there can not exist a general formula, only numerical approximatio ...
... In elementary algebra, quadratic formula are given for solving all second degree polynomial equations in one variable. There are also formulae for the cubic and quartic equations. For higher degrees, Abel–Ruffini theorem asserts that there can not exist a general formula, only numerical approximatio ...
§33 Polynomial Rings
... essential point is that a polynomial can be written as a sum of monomials, and a monomial is determined as soon as the index r and the possibly nonzero element a is given. We can choose other notations for monomials, of course, as long as they display the index r and the possibly nonzero element a. ...
... essential point is that a polynomial can be written as a sum of monomials, and a monomial is determined as soon as the index r and the possibly nonzero element a is given. We can choose other notations for monomials, of course, as long as they display the index r and the possibly nonzero element a. ...
FINITE FIELDS OF THE FORM GF(p)
... Let set S of polynomial coefficients is a finite field Zp, and polynomials have degree from 0 to n-1. There are totally pn different such polynomials. The definition consists of the following elements: 1. Arithmetic follows the ordinary rules of polynomial arithmetic using the basic rules of algebra ...
... Let set S of polynomial coefficients is a finite field Zp, and polynomials have degree from 0 to n-1. There are totally pn different such polynomials. The definition consists of the following elements: 1. Arithmetic follows the ordinary rules of polynomial arithmetic using the basic rules of algebra ...
FINITE FIELDS OF THE FORM GF(p)
... Let set S of polynomial coefficients is a finite field Zp, and polynomials have degree from 0 to n-1. There are totally pn different such polynomials. The definition consists of the following elements: 1. Arithmetic follows the ordinary rules of polynomial arithmetic using the basic rules of algebra ...
... Let set S of polynomial coefficients is a finite field Zp, and polynomials have degree from 0 to n-1. There are totally pn different such polynomials. The definition consists of the following elements: 1. Arithmetic follows the ordinary rules of polynomial arithmetic using the basic rules of algebra ...
On the multiplicity of zeroes of polyno
... is a direct consequence of Theorem 3.3 in [1], which we repeat here for the sake of completeness (but see also [5] for this same statement in the case of polynomials). Theorem 1.6 (Zeroes of a regular product for power series). Let f, g be given quaternionic power series with radii greater than R an ...
... is a direct consequence of Theorem 3.3 in [1], which we repeat here for the sake of completeness (but see also [5] for this same statement in the case of polynomials). Theorem 1.6 (Zeroes of a regular product for power series). Let f, g be given quaternionic power series with radii greater than R an ...
Polynomials
... fields and the axioms they satisfy, you can visit Wikipedia’s site and read the article on fields: http: //en.wikipedia.org/wiki/Field_%28mathematics%29. While you are there, you can also read the article on Galois Fields and learn more about some of their applications and elegant properties which w ...
... fields and the axioms they satisfy, you can visit Wikipedia’s site and read the article on fields: http: //en.wikipedia.org/wiki/Field_%28mathematics%29. While you are there, you can also read the article on Galois Fields and learn more about some of their applications and elegant properties which w ...
s13 - Math-UMN
... First, by Eisenstein’s criterion, x5 −2 is irreducible over Q, so the fifth root of 2 generates a quintic extension of Q. Certainly a fifth root of 4 lies in such an extension, so must be either rational or generate the quintic extension, by multiplicativity of field extension degrees in towers. Sin ...
... First, by Eisenstein’s criterion, x5 −2 is irreducible over Q, so the fifth root of 2 generates a quintic extension of Q. Certainly a fifth root of 4 lies in such an extension, so must be either rational or generate the quintic extension, by multiplicativity of field extension degrees in towers. Sin ...