19. Rings, Fields and Polynomials
The lecture focuses on rings, fields, and polynomials over rings as algebraic structures. Key properties of rings and fields are examined, including their axioms and examples, particularly involving integers modulo N. The discussion extends to polynomials defined over rings, detailing operations like addition and multiplication and their adherence to ring axioms. Throughout, the significance of invertible elements and their implications for these algebraic structures are highlighted.
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Sections
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What we have learnt
- Rings are algebraic structures defined by a set and two operations satisfying specific axioms.
- Fields are a special type of ring where every non-zero element is invertible.
- Polynomials can be defined over rings and must adhere to the operations and properties established by their corresponding rings.
Key Concepts
- -- Ring
- An algebraic structure consisting of a set equipped with two operations satisfying ring axioms, including closure, associativity, and distributivity.
- -- Field
- A set with two operations that satisfies field axioms; every non-zero element has a multiplicative inverse.
- -- Polynomial
- An expression involving variables raised to non-negative integer powers and coefficients from a ring, following specific operations of addition and multiplication.
Additional Learning Materials
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