Finite fields are like secret codes for math nerds. They're special number systems with a limited set of elements, always a power of a prime number. Understanding their structure and properties is key to unlocking their power in cryptography and coding theory.

In this section, we'll dive into the nitty-gritty of finite fields. We'll explore how they're built, their unique characteristics, and the cool tricks you can do with their arithmetic. Get ready to level up your finite field game!

Finite Fields and Characteristics

Definition and Properties

• A finite field is a field that contains a finite number of elements, also known as a Galois field
• The order (number of elements) of a finite field is always a prime power, denoted as $q = p^n$, where $p$ is a prime number and $n$ is a positive integer
• There exists a unique finite field (up to isomorphism) for each prime power order $q = p^n$, denoted as $GF(q)$ or $F_q$

Characteristic of a Finite Field

• The characteristic of a finite field is the smallest positive integer $m$ such that $m 1 = 0$, where $1$ is the multiplicative identity of the field
• In a finite field of order $q = p^n$, the characteristic is always the prime $p$
  • For example, in the finite field $GF(2^3)$, the characteristic is $2$
  • This means that for any element $a$ in $GF(2^3)$, $2 a = 0$

Constructing Finite Fields

Quotient Rings and Irreducible Polynomials

• Finite fields can be constructed as quotient rings of polynomial rings over the integers modulo a prime
• An irreducible polynomial is a polynomial that cannot be factored into the product of two non-constant polynomials over a given field
• To construct a finite field of order $q = p^n$, find an irreducible polynomial $f(x)$ of degree $n$ over the field $Z_p$ (integers modulo $p$)

Elements and Arithmetic Operations

• The elements of the finite field are the polynomials of degree less than $n$ with coefficients in $Z_p$, with arithmetic operations performed modulo $f(x)$
  • For example, in $GF(2^2)$ constructed using the irreducible polynomial $f(x) = x^2 + x + 1$, the elements are ${0, 1, x, x+1}$
• The multiplicative group of a finite field is cyclic, generated by a primitive element
  • A primitive element is a generator of the multiplicative group, meaning that every non-zero element can be expressed as a power of the primitive element

Arithmetic in Finite Fields

Addition and Subtraction

• Addition in a finite field is performed by adding the coefficients of the corresponding terms of the polynomials modulo the characteristic $p$
  • For example, in $GF(2^2)$, $(x + 1) + (x) = 1$ because $1 + 1 \equiv 0 \pmod{2}$
• Subtraction in a finite field is performed by subtracting the coefficients of the corresponding terms of the polynomials modulo the characteristic $p$
  • For example, in $GF(2^2)$, $(x + 1) - (x) = 1$ because $1 - 1 \equiv 0 \pmod{2}$

Multiplication and Division

• Multiplication in a finite field is performed by multiplying the polynomials and then reducing the result modulo the irreducible polynomial $f(x)$ and the characteristic $p$
  • For example, in $GF(2^2)$ with $f(x) = x^2 + x + 1$, $(x + 1) (x) = x^2 + x \equiv 1 \pmod{f(x)}$
• Division in a finite field is performed by multiplying the dividend by the inverse of the divisor, where the inverse is computed using the extended Euclidean algorithm
  • For example, in $GF(2^2)$ with $f(x) = x^2 + x + 1$, $(x + 1) / (x) = (x + 1) (x + 1) \equiv 1 \pmod{f(x)}$ because $(x + 1)$ is the inverse of $(x)$

Properties of Finite Fields

Primitive Elements

• Every finite field has a primitive element, which is a generator of the multiplicative group of the field
• The number of primitive elements in a finite field of order $q$ is $\phi(q-1)$, where $\phi$ is Euler's totient function
  • For example, in $GF(2^3)$, there are $\phi(7) = 6$ primitive elements
• Primitive elements are used in various applications, such as constructing pseudo-random number generators and designing error-correcting codes

Trace and Norm Functions

• The trace and norm functions are important maps between finite fields and their subfields, which can be used to analyze the structure and properties of the fields
  • The trace function $Tr_{F_q/F_p}(a)$ of an element $a$ in $F_q$ is the sum of its conjugates over the subfield $F_p$
  • The norm function $N_{F_q/F_p}(a)$ of an element $a$ in $F_q$ is the product of its conjugates over the subfield $F_p$
• These functions have various properties, such as linearity and multiplicativity, which can be used to solve equations and study the structure of finite fields