🏃🏽‍♀️Galois Theory Unit 10 – Finite Fields and Their Applications

Definition and Basics

• A finite field is a field that contains a finite number of elements
• Finite fields are also known as Galois fields, named after the mathematician Évariste Galois
• The order of a finite field is the number of elements it contains and is always a prime power, denoted as $q = p^n$ where $p$ is a prime number and $n$ is a positive integer
• Finite fields are unique up to isomorphism for a given order, meaning any two finite fields of the same order have the same structure and properties
• The elements of a finite field can be represented as polynomials over the prime field $\mathbb{F}_p$, where the coefficients are taken modulo $p$
  • For example, the finite field of order 4, denoted as $\mathbb{F}_4$, can be represented as polynomials over $\mathbb{F}_2$ (coefficients are 0 or 1)
• Finite fields are important in various applications, including cryptography, coding theory, and combinatorics

Field Characteristics and Properties

• The characteristic of a finite field is the smallest positive integer $p$ such that $p \cdot 1 = 0$, where $1$ is the multiplicative identity of the field
  • In a finite field of order $q = p^n$, the characteristic is always the prime number $p$
• Finite fields satisfy all the axioms of a field: closure, associativity, commutativity, distributivity, identity elements, and inverses for both addition and multiplication
• Every non-zero element in a finite field has a multiplicative inverse, making the field a division ring
• The multiplicative group of a finite field is cyclic, meaning there exists a primitive element that generates all non-zero elements of the field
• Fermat's Little Theorem holds in finite fields: for any non-zero element $a$ in a finite field of characteristic $p$, $a^{p-1} = 1$
• Finite fields have no proper infinite subfields, and every subfield of a finite field is also finite

Finite Field Construction

• Finite fields can be constructed using irreducible polynomials over the prime field $\mathbb{F}_p$
• An irreducible polynomial is a polynomial that cannot be factored into non-constant polynomials of lower degree over the given field
• To construct a finite field of order $q = p^n$, find an irreducible polynomial $f(x)$ of degree $n$ over $\mathbb{F}_p$
  • The elements of the finite field are then the polynomials of degree less than $n$ with coefficients in $\mathbb{F}_p$, and arithmetic is performed modulo $f(x)$
• For example, to construct $\mathbb{F}_8$, use the irreducible polynomial $f(x) = x^3 + x + 1$ over $\mathbb{F}_2$
  • The elements of $\mathbb{F}_8$ are the polynomials $\{0, 1, x, x+1, x^2, x^2+1, x^2+x, x^2+x+1\}$ with arithmetic performed modulo $f(x)$
• Finite fields can also be constructed using other methods, such as adjoining roots of irreducible polynomials or using Zech logarithms

Polynomial Representation

• Elements of a finite field can be represented as polynomials over the prime field $\mathbb{F}_p$
• Polynomial arithmetic in finite fields is performed modulo the characteristic $p$ and modulo an irreducible polynomial $f(x)$ of degree $n$
• Addition of polynomials in a finite field is done by adding the coefficients of like terms modulo $p$
• Multiplication of polynomials in a finite field is done by multiplying the polynomials as usual and then reducing the result modulo $f(x)$ and modulo $p$
  • For example, in $\mathbb{F}_8$ with $f(x) = x^3 + x + 1$, multiplying $x^2 + 1$ by $x + 1$ gives $(x^2 + 1)(x + 1) = x^3 + x^2 + x + 1 \equiv x^2 + x$ modulo $f(x)$ and modulo 2
• Division of polynomials in a finite field is done by finding the multiplicative inverse of the divisor and then multiplying by the dividend
• Polynomial factorization over finite fields is an important problem with applications in cryptography and coding theory

Field Extensions and Subfields

• A field extension is a field that contains a given base field as a subfield
• For finite fields, every extension is a finite extension, meaning the extension field is a finite-dimensional vector space over the base field
• The degree of a field extension is the dimension of the extension field as a vector space over the base field
• A field extension of degree $n$ over a finite field $\mathbb{F}_q$ has order $q^n$
• Every finite field has a unique subfield of order $p$, where $p$ is the characteristic of the field, called the prime subfield
• Other subfields of a finite field can be found by considering the divisors of the degree of the extension
  • For example, $\mathbb{F}_{2^6}$ has subfields of order 2, 4, and 8 (corresponding to the divisors of 6)
• The Galois group of a finite field extension is always cyclic and generated by the Frobenius automorphism, which maps each element to its $p$-th power

Algebraic Closure

• An algebraically closed field is a field in which every non-constant polynomial has a root
• Finite fields are not algebraically closed, as there exist non-constant polynomials without roots in the field
  • For example, the polynomial $x^2 + 1$ has no roots in $\mathbb{F}_3$
• The algebraic closure of a finite field $\mathbb{F}_q$ is an infinite field containing $\mathbb{F}_q$ as a subfield, denoted as $\overline{\mathbb{F}}_q$
• The algebraic closure of a finite field is the union of all finite extensions of the field, i.e., $\overline{\mathbb{F}}_q = \bigcup_{n=1}^{\infty} \mathbb{F}_{q^n}$
• Every finite field is perfect, meaning every algebraic extension is separable
• The absolute Galois group of a finite field (the Galois group of its algebraic closure) is isomorphic to the profinite completion of the integers, $\hat{\mathbb{Z}}$

Applications in Cryptography

• Finite fields are widely used in cryptography due to their algebraic properties and the difficulty of solving certain problems, such as the discrete logarithm problem
• The discrete logarithm problem in a finite field is the problem of finding an integer $k$ such that $g^k = h$ for given elements $g$ and $h$, which is believed to be computationally infeasible for large field orders
• Elliptic curve cryptography (ECC) uses the group of points on an elliptic curve over a finite field to construct public-key cryptosystems
  • ECC offers similar security to traditional systems (like RSA) with smaller key sizes, making it more efficient
• The Advanced Encryption Standard (AES) uses arithmetic in the finite field $\mathbb{F}_{2^8}$ as part of its encryption and decryption processes
• Finite fields are also used in the construction of various cryptographic primitives, such as hash functions, digital signatures, and key exchange protocols

Advanced Concepts and Open Problems

• Finite fields have connections to various areas of mathematics, including number theory, algebraic geometry, and combinatorics
• The Weil conjectures, proved by André Weil in the 1940s, relate the number of points on algebraic varieties over finite fields to the topology of the varieties over the complex numbers
• The Riemann hypothesis for finite fields, proved by André Weil in 1948, is an analog of the classical Riemann hypothesis and concerns the zeros of zeta functions associated with algebraic curves over finite fields
• Finite fields are used in the construction of various combinatorial objects, such as linear codes, projective planes, and difference sets
• Open problems related to finite fields include:
  • The classification of irreducible polynomials over finite fields
  • The distribution of primitive elements in finite fields
  • The construction of finite fields with specific properties (e.g., large multiplicative subgroups)
• Research in finite fields continues to find new applications and connections to other areas of mathematics and computer science