Discrete Mathematics - Vol 3 | 19. Rings, Fields and Polynomials by Abraham | Learn Smarter
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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

  • 19.1

    Discrete Mathematics

    This section covers the concepts of rings, fields, and polynomials in discrete mathematics, outlining their definitions, properties, and examples.

    Learning Practice

  • 19.2

    Rings, Fields And Polynomials

    This section discusses the definitions and key properties of rings, fields, and polynomials over rings.

    Learning Practice

  • 19.2.1

    Definition Of A Ring

    In this section, the concept of a ring in abstract algebra is introduced, detailing its structure, operations, and essential axioms.

    Learning Practice

  • 19.2.2

    Axioms Of A Ring

    This section discusses the properties that define a ring in abstract algebra, including the essential axioms necessary for a set and operations to qualify as a ring.

    Learning Practice

  • 19.2.3

    Examples Of Rings

    This section introduces rings, detailing their axioms and specific examples, including the ring of integers modulo N, and explores invertible elements within rings.

    Learning Practice

  • 19.2.4

    Invertible Elements Of A Ring

    This section discusses the concept of invertible elements within the structure of a ring, detailing which elements have multiplicative inverses and introducing the set of all invertible elements, U(ℝ).

    Learning Practice

  • 19.2.5

    The Set U(ℝ)

    This section defines the set of invertible elements in a ring and establishes its significance in relation to ring theory.

    Learning Practice

  • 19.2.6

    Proof Of Invertible Elements Forming A Subgroup

    This section explains the concept of invertible elements in a ring and proves that the set of such elements forms a subgroup.

    Learning Practice

  • 19.2.7

    Definition Of A Field

    This section defines fields in abstract algebra, detailing their axioms and significance in algebraic structures.

    Learning Practice

  • 19.2.8

    Field Axioms

    This section discusses the axioms defining fields, emphasizing the relational structure of sets under addition and multiplication, and contrasts these with rings.

    Learning Practice

  • 19.2.9

    Polynomials Over Rings

    This section provides an understanding of polynomials defined over rings and their operations, drawing parallels with familiar polynomial concepts.

    Learning Practice

References

ch67.pdf

Class Notes

Memorization

What we have learnt

  • Rings are algebraic structu...
  • Fields are a special type o...
  • Polynomials can be defined ...

Final Test

Revision Tests