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A group is a mathematical structure consisting of a set of elements together with an operation that satisfies four properties:

  1. Closure: For any two elements

    and  in the group, the result of combining them using the group operation is also an element of the group.

  2. Associativity: The group operation is associative, meaning that for any elements

    , and in the group, 

  3. Identity Element: There exists an element

    in the group such that for any element


  4. Inverse Element: For every element 

    (called the inverse of 

    ) such that

    , where 

    is the identity element.


A field is a set equipped with two operations, addition and multiplication, satisfying the following properties:

  1. Closure: Addition and multiplication of any two elements in the field result in another element of the field.
  2. Associativity: Addition and multiplication are associative operations.
  3. Commutativity: Both addition and multiplication are commutative, meaning


    and 𝑎

    for any elements 

    in the field.

  4. Identity Elements: There exist additive and multiplicative identity elements

    and 1

    , respectively, such that 𝑎+0=𝑎

    and 𝑎⋅1=𝑎

    for any element 

    in the field.

  5. Inverse Elements: Every nonzero element in the field has an additive and multiplicative inverse. For any element

    in the field, its additive inverse is denoted as

    , and its multiplicative inverse is denoted as 

    . Note that

    does not have a multiplicative inverse.

  6. Distributive Property: Multiplication distributes over addition, meaning


    for any elements 


    in the field.

Finite Field of the Form GF(p)

A finite field of the form GF(p), where

is a prime number, consists of integers modulo

under addition and multiplication. It is denoted as




. The elements of GF(p) are integers in the range



Properties of GF(p):

  • Size: GF(p) has elements.
  • Addition: Addition in GF(p) is performed modulo 

    , then 𝑎+b−𝑝

    is taken as the result.

  • Multiplication: Multiplication in GF(p) is performed modulo

    , meaning that if 

    , then 

    is taken as the result.

  • Additive and Multiplicative Inverses: Every nonzero element

    in GF(p) has an additive inverse

    and a multiplicative inverse 

    such that

    and 𝑎⋅𝑎−1=1



  • Closure, Associativity, Commutativity: These properties hold for addition and multiplication in GF(p) as they do for any field.

Finite fields of the form GF(p) are widely used in cryptography, error-correcting codes, and various other areas of mathematics and computer science due to their algebraic properties and computational efficiency.