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EECS-1019c: Assignment #7
EECS-1019c: Assignment #7

CM222, Linear Algebra Mock Test 3 Solutions 1. Let P2 denote the
CM222, Linear Algebra Mock Test 3 Solutions 1. Let P2 denote the

... Solution: a), c) Explanation: a) is the definition. c) is the same as a) since the eigenvalues with eigenvalue λ are precisely the non-zero vectors in the null space of A − λI. b) is incorrect, since for example the geometric multiplicity of the eigenvalue 1 for the 2 × 2 identity matrix is 2, but t ...
Euclidean algorithm
Euclidean algorithm

Logarithms in running time
Logarithms in running time

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Full text

TRUE/FALSE. Write `T` if the statement is true and `F` if the
TRUE/FALSE. Write `T` if the statement is true and `F` if the

... 10) A field is a set in which we can do addition, subtraction, multiplication and division without leaving the set. ...
enumerating polynomials over finite fields
enumerating polynomials over finite fields

... A polynomial is irreducible if it is of positive degree and cannot be factored into polynomials of strictly smaller degree. So for instance every polynomial of degree one is irreducible. In fact, every polynomial can be uniquely factored into irreducible polynomials (possibly repeated). More precise ...
CHAP11 Z2 Polynomials
CHAP11 Z2 Polynomials

... Cast your mind back to the time when you first learnt about complex numbers. Your whole world of numbers was the field of real numbers. There were many polynomial equations which had no solutions such as x2 + 1 = 0. What we did was to invent solutions for this polynomial. A new “imaginary” number “i ...
1 PROBLEM SET 9 DUE: May 5 Problem 1(algebraic integers) Let K
1 PROBLEM SET 9 DUE: May 5 Problem 1(algebraic integers) Let K

... (3). Let m be a maximal ideal of OK , and let k = OK /m be the residue field. Verify there exists a natural ring epimorphism OK [x] → k[x]. (4). (Eisenstein’s criterion) Let f (x) = an xn +an−1 xn−1 +....+a0 ∈ Z[x], satisfying an 6≡ 0 mod p, ai ≡ 0 mod p f or i 6= n, a0 6≡ 0 mod p2 , where p is a pr ...
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Assignment Sheet (new window)

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Lesson4 - Purdue Math

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Grade 9 Math Unit 3 Patterns and Relationships Part One

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3.3 Introduction to Polynomials

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MatlabTutorial

... functions. To use these tools, the polynomial should be represented as a vector with the leftmost number being the highest power and the rightmost number being the constant. For example, x² + 2x + 1 would be represented as [1 2 1]. • The roots function gives the roots of the polynomial and polyval e ...
Review Problems
Review Problems

... 4. Factor x5 − 2x4 − 2x3 + 12x2 − 15x − 2 over Q. Solution: The possible rational roots are ±1, ±2, and since 2 is a root we have the factorization x5 − 2x4 − 2x3 + 12x2 − 15x − 2 = (x − 2)(x4 − 2x2 + 8x + 1). The only possible rational roots of the second factor are 1 and −1, and these do not work. ...
Final Exam conceptual review
Final Exam conceptual review

... (b) Goals: Determine the number of elements of F [x]/hpi, find all units and all zero divisors of F [x]/hpi (c) Homework problems: #10 (d) Additional practice problems: #9, 11 Proof-based problems. Here is a collection of additional (some more challenging) proofs to practice with that require knowle ...
43. Here is the picture: • • • • • • • • • • • • •
43. Here is the picture: • • • • • • • • • • • • •

... 44. Because the polynomial is primitive, it is irreducible over Z if and only if it is irreducible over Q. As it has degree 3, it is irreducible over Q if and only if it has no roots in Q. Moreover, if a = n/d ∈ Q is a root, then n|4 and d|1, so a ∈ {±1, ±2, ±4}. Checking these one by one, we find t ...
Unit 9 – Polynomials Algebra I Essential Questions Enduring
Unit 9 – Polynomials Algebra I Essential Questions Enduring

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3 Evaluation, Interpolation and Multiplication of Polynomials

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Chapter 5 Review

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Wedderburn`s Theorem on Division Rings: A finite division ring is a

... Φd (x) where either d|mi or d = n; so the quotient is a product of the Φd (x)’s where d is a proper divisor of n that does not divide mi ; hence the quotient is a polynomial with integer coefficients. Substituting the integer q for the variable x, we see that the integer Φn (q) divides the integer ( ...
Principal Ideal Domains
Principal Ideal Domains

... Proof. The result follows from Theorems 57 and 62. Theorem 63. Every nonzero prime ideal in a PID is a maximal ideal. Corollary 64. If R is a commutative ring such that the polynomial ring R[x] is a PID, then R is necessarily a field. Example 65. Here are a few quick examples. (1) We already know th ...
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PDF

... The polynomial ring over R in two variables X, Y is defined to be R[X, Y ] := R[X][Y ] ∼ = R[Y ][X]. Elements of R[X, Y ] are called polynomials in the indeterminates X and Y with coefficients in R. A monomial in R[X, Y ] is a polynomial which is simultaneously a monomial in both X and Y , when cons ...
The Rational Numbers - Stony Brook Mathematics
The Rational Numbers - Stony Brook Mathematics

< 1 ... 38 39 40 41 42 43 44 45 >

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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