Group

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
   

   $a$ and $b$ 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
   

   $a$, $b$, and $c$ in the group, 

   
   $(a⋅b)⋅c=a⋅(b⋅c)$

3. Identity Element: There exists an element

   $$

   e in the group such that for any element
   $𝑎$

   a⋅e=e⋅a=a.

4. Inverse Element: For every element 
   $$a𝑎−1

   a−1 (called the inverse of 

   
   $$

   a) such that
   $𝑎\cdot {𝑎}^{-1}={𝑎}^{-1}\cdot 𝑎=𝑒$

   a⋅a−1=a−1⋅a=e, where 

   
   $$

   e is the identity element.

Field

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

   $𝑎+𝑏=𝑏+𝑎$

   a+b=b+a and 𝑎
   $\cdot 𝑏=𝑏\cdot 𝑎$

   
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   $$

   
   $\mathrm{<span\; class="math\; math-inline"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="katex"><span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal">b</span></span></span></span></span><span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;">\; in\; the\; field.</span>}$

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

   $0$ and 1

   $1$, respectively, such that 𝑎+0=𝑎

   $a+0=a$ and 𝑎⋅1=𝑎

   $a\cdot 1=a$ for any element 

   
   $$

   $a$ in the field.

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

   $$

   a in the field, its additive inverse is denoted as
   $-𝑎$

   −a, and its multiplicative inverse is denoted as 

   
   ${}^{}$

   a−1. Note that
   $0$

   0 does not have a multiplicative inverse.

6. Distributive Property: Multiplication distributes over addition, meaning

   $𝑎\cdot (𝑏+𝑐)=(𝑎\cdot 𝑏)+(𝑎\cdot 𝑐)$

   a⋅(b+c)=(a⋅b)+(a⋅c) for any elements 

   
   $$

   a,
   $$

   c in the field.

Finite Field of the Form GF(p)

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

p is a prime number, consists of integers modulo

p under addition and multiplication. It is denoted as

$\text{GF}(𝑝)$

GF(p) or

${\mathbb{𝐹}}_{𝑝}$

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

$[0,𝑝-1]$

[0,p−1].

Properties of GF(p):

• Size: GF(p) has $p$ elements.
• Addition: Addition in GF(p) is performed modulo 
  $$p𝑎+𝑏≥𝑝

  a+b≥p, then 𝑎+b−𝑝

  a+b−p is taken as the result.

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

  $p$, meaning that if 

  $a\cdot b\ge p$, then 

  $(a\cdot b)\mathrm{mod}p$ is taken as the result.

• Additive and Multiplicative Inverses: Every nonzero element

  $$

  a in GF(p) has an additive inverse
  $-𝑎$

  −a and a multiplicative inverse 

  
  ${}^{}$

  a−1 such that
  $𝑎+(-𝑎)=0$

  a+(−a)=0 and 𝑎⋅𝑎−1=1

  a⋅a−1=1 modulo 

  p.

• 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.