Discrete Mathematics - Vol 3 | Overview 41 by Abraham | Learn Smarter
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Overview 41

The lecture discusses the properties of finite fields, particularly the order of a finite field, which can be expressed as a prime number raised to a power. A strong theorem is proven, showing that for any finite field with prime characteristic, the number of elements within the field is of the form p^r. Additionally, the process of constructing finite fields using irreducible monic polynomials is explored, providing a framework to follow for finite fields of various orders.

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Sections

  • 1

    Finite Fields And Properties Ii

    This section discusses finite fields, specifically their order, characterizing properties, and the construction of finite fields using irreducible polynomials.

    Learning Practice

  • 1.1

    Order Of A Finite Field

    This section discusses the order of a finite field, which is defined as the number of elements in the field, and explores key properties related to this concept.

    Learning Practice

  • 1.2

    Properties Of The Order Of A Finite Field

    This section outlines the definition and properties of the order of finite fields, showing that the number of elements in such a field can be expressed as a prime raised to an integer exponent.

    Learning Practice

  • 1.3

    Proof Of The Theorem

    This section discusses the proof of a theorem related to the order of finite fields, demonstrating that the number of elements in such fields is of the form p^r where p is a prime number.

    Learning Practice

  • 1.4

    Span Of The Field

    This section discusses the concept of the span of a finite field and its properties.

    Learning Practice

  • 1.5

    Minimal Spanning Set

    This section explores the concept of minimal spanning sets in finite fields, detailing how to identify a collection of elements that are essential for spanning the entire field.

    Learning Practice

  • 1.6

    Mapping From ℤ^r To Field F

    This section explores the properties of finite fields, specifically the mapping from ℤ^r to a finite field F, highlighting the order of the field and the characteristics of finite fields.

    Learning Practice

  • 1.7

    Proof Of Bijection

    This section explores the proof of the number of elements in a finite field being of the form pr, where p is a prime number.

    Learning Practice

  • 1.8

    Construction Of Finite Fields

    This section discusses the construction of finite fields, focusing on their order and the properties associated with them.

    Learning Practice

  • 1.9

    Existence Of Irreducible Polynomials

    This section discusses the existence of irreducible polynomials within finite fields and demonstrates how these polynomials are crucial for constructing finite fields of a specific order.

    Learning Practice

  • 1.10

    Field Definition And Operations

    This section defines the properties of finite fields, focusing on their cardinality and span.

    Learning Practice

  • 1.11

    Existence Of Multiplicative Inverse

    This section discusses the existence and properties of multiplicative inverses in finite fields, emphasizing their significance and role in linear combinations.

    Learning Practice

  • 1.12

    Final Construction Of Fields

    This section discusses the order of finite fields and their construction through polynomials, emphasizing their cardinality.

    Learning Practice

  • 1.13

    Examples Of Field Construction

    This section elaborates on the construction of finite fields, detailing their order and properties, particularly focusing on examples using irreducible polynomials.

    Learning Practice

References

ch70.pdf

Class Notes

Memorization

What we have learnt

  • The order of a finite field...
  • Finite fields can be constr...
  • The mapping between r-tuple...

Final Test

Revision Tests