Discrete Mathematics - Vol 3 | 22. Finite Fields and Properties I by Abraham | Learn Smarter
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22. Finite Fields and Properties I

The chapter discusses the construction of finite fields and their properties, specifically focusing on the characteristic of a field. Through various examples, it illustrates how finite fields operate under addition and multiplication modulo an irreducible polynomial, establishing essential concepts such as field axioms, cyclic groups, and the significance of prime characteristics. Additionally, it proves that the characteristic of any finite field is always a prime number.

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Sections

  • 22

    Finite Fields And Properties I

    This section discusses the construction and properties of finite fields, focusing on their characteristic and the verification of field axioms.

    Learning Practice

  • 22.1

    Construction Of Finite Field With 9 Elements

    This section covers the construction of a finite field with 9 elements by defining polynomial operations over a specified set.

    Learning Practice

  • 22.2

    Verification Of Field Axioms

    The section discusses finite fields, specifically focusing on their construction, properties, and verification of field axioms.

    Learning Practice

  • 22.3

    Characteristic Of A Field

    This section discusses finite fields and defines the characteristic of a field, explaining how to determine it and its significance.

    Learning Practice

  • 22.3.1

    Examples Of Characteristic Of A Field

    This section discusses the concept of the characteristic of a field, especially in the context of finite fields, including definitions, examples, and properties.

    Learning Practice

  • 22.3.2

    Theorem On Characteristic Of Finite Fields

    This section explores the characteristic of finite fields, highlighting its importance and properties.

    Learning Practice

  • 22.3.2.1

    Proof By Contradiction

    This section explores the concept of proof by contradiction, specifically relating to the characteristic of finite fields, demonstrating that the characteristic must be a prime number through a contradiction approach.

    Learning Practice

References

ch69.pdf

Class Notes

Memorization

What we have learnt

  • The construction of finite ...
  • Each non-zero element in a ...
  • The characteristic of a fin...

Final Test

Revision Tests