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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.
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Sections
References
ch68 - part A.pdfClass Notes
Memorization
What we have learnt
- Polynomials over fields exh...
- The concept of division of ...
- Irreducibility indicates th...
Final Test
Revision Tests
What we have learnt
- 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.
Key Concepts
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Term: Polynomials over Fields
Definition: These are polynomials where the coefficients belong to a field, allowing for unique characteristics in operations like addition, multiplication, and division.
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Term: GCD of Polynomials
Definition: The greatest common divisor of two polynomials, which generalizes the concept from integers, indicating the largest polynomial that divides both without a remainder.
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Term: Irreducible Polynomial
Definition: A non-constant polynomial that cannot be factored into the product of two non-constant polynomials.
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Term: Factor Theorem
Definition: If a polynomial f(x) equals 0 when x is substituted with α, then (x - α) is a factor of f(x).