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20. Polynomials Over Fields and Properties
The chapter elaborates on polynomials over fields, detailing their properties, division, and factorization. It introduces the concepts of irreducible and reducible polynomials, and explores the GCD of polynomials along with the Euclidean algorithm for finding it. Additionally, the chapter presents the factor theorem and concludes with discussions on polynomial factorization.
Sections
This section discusses the properties of polynomials over fields, including division, factorization, and the concepts of reducible and irreducible polynomials.
Polynomials over fields exhibit unique properties that differ from those over rings.
The concept of division of polynomials mirrors traditional integer division but introduces unique structural elements for polynomials over fields.
Irreducibility indicates that a polynomial cannot be factored into non-trivial factors, playing a key role in polynomial theory.
Polynomials over Fields
These are polynomials where the coefficients belong to a field, allowing for unique characteristics in operations like addition, multiplication, and division.
GCD of Polynomials
The greatest common divisor of two polynomials, which generalizes the concept from integers, indicating the largest polynomial that divides both without a remainder.
Irreducible Polynomial
A non-constant polynomial that cannot be factored into the product of two non-constant polynomials.
Factor Theorem
If a polynomial f(x) equals 0 when x is substituted with α, then (x - α) is a factor of f(x).
Practice Exercises
Total Questions
2
Estimated Time
4 min
Passing Score
70%
Instructions
- Read each question carefully
- You can use hints if you need help
- Complete all questions before submitting
1 more question available
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