Galois Theory - Basic Elements, Finite Fields

Introduction to Galois Theory

Galois theory is a branch of abstract algebra that connects field theory with group theory. It was developed by Évariste Galois in the 19th century. The primary goal of Galois theory is to understand the solvability of polynomial equations by radicals. This involves studying the symmetries of the roots of a polynomial, which are captured by a specific group called the Galois group.

At its core, Galois theory establishes a fundamental correspondence between the intermediate fields of a field extension and the subgroups of its Galois group. This correspondence provides powerful tools to analyze the structure of field extensions and, consequently, the properties of polynomial equations.

Field Extensions

A field extension is a pair of fields $(K, F)$ such that $F$ is a subfield of $K$. We denote this as $K/F$. For example, the field of complex numbers $\mathbb{C}$ is an extension of the field of real numbers $\mathbb{R}$, denoted $\mathbb{C}/\mathbb{R}$. The field $\mathbb{Q}(\sqrt{2}) = \{a + b\sqrt{2} \mid a, b \in \mathbb{Q}\}$ is an extension of the field of rational numbers $\mathbb{Q}$.

The degree of a field extension $K/F$, denoted $[K:F]$, is the dimension of $K$ as a vector space over $F$. If $[K:F]$ is finite, we say $K/F$ is a finite extension. A key property is the tower law: if $L/K$ and $K/F$ are finite extensions, then $L/F$ is also a finite extension, and $[L:F] = [L:K][K:F]$.

An element $\alpha$ is algebraic over a field $F$ if it is a root of a non-zero polynomial with coefficients in $F$. The smallest field containing $F$ and $\alpha$ is denoted $F(\alpha)$. The extension $F(\alpha)/F$ is finite if and only if $\alpha$ is algebraic over $F$. The degree of this extension, $[F(\alpha):F]$, is the degree of the minimal polynomial of $\alpha$ over $F$.

Splitting Fields

Given a field $F$ and a polynomial $p(x) \in F[x]$, a splitting field for $p(x)$ over $F$ is a field extension $K$ of $F$ such that $p(x)$ factors into linear factors in $K[x]$, and $K$ is the smallest such field. In other words, $p(x) = c(x-\alpha_1)\dots(x-\alpha_n)$ for some $c \in F$ and $\alpha_i \in K$, and $K = F(\alpha_1, \dots, \alpha_n)$.

Every polynomial over a field has a splitting field. If $p(x)$ is irreducible over $F$, then any two splitting fields of $p(x)$ over $F$ are isomorphic over $F$.

For example, the splitting field of $x^2 - 2$ over $\mathbb{Q}$ is $\mathbb{Q}(\sqrt{2})$. The splitting field of $x^3 - 2$ over $\mathbb{Q}$ is $\mathbb{Q}(\sqrt[3]{2}, \omega)$, where $\omega = e^{2\pi i/3}$ is a primitive cube root of unity. This field contains all three roots: $\sqrt[3]{2}$, $\sqrt[3]{2}\omega$, and $\sqrt[3]{2}\omega^2$.

Normal and Separable Extensions

A finite extension $K/F$ is called normal if it is the splitting field of some polynomial in $F[x]$. Equivalently, $K/F$ is normal if every irreducible polynomial in $F[x]$ that has a root in $K$ splits completely into linear factors in $K[x]$.

A finite extension $K/F$ is called separable if for every element $\alpha \in K$, the minimal polynomial of $\alpha$ over $F$ has distinct roots. If $F$ has characteristic zero (like $\mathbb{Q}$ or $\mathbb{R}$), all algebraic extensions are separable. For fields of positive characteristic, separability requires careful consideration.

A Galois extension is a finite, normal, and separable field extension.

Galois Groups

Let $K/F$ be a Galois extension. The Galois group of $K$ over $F$, denoted $\text{Gal}(K/F)$, is the group of field automorphisms of $K$ that fix every element of $F$. An automorphism $\sigma: K \to K$ is an isomorphism from $K$ to itself. If $\sigma$ fixes $F$, it means $\sigma(a) = a$ for all $a \in F$.

The operation in the Galois group is the composition of automorphisms. The identity map is the identity element of the group.

A fundamental theorem of Galois theory states that if $K/F$ is a Galois extension, then the order of the Galois group is equal to the degree of the extension: $|\text{Gal}(K/F)| = [K:F]$.

Example: Consider the extension $\mathbb{Q}(i)/\mathbb{Q}$. The complex conjugate map $\sigma(a+bi) = a-bi$ is an automorphism. It fixes $\mathbb{Q}$ since $a, b \in \mathbb{Q}$. The roots of $x^2+1$ are $i$ and $-i$. The splitting field is $\mathbb{Q}(i)$. The degree is $[\mathbb{Q}(i):\mathbb{Q}] = 2$. The Galois group $\text{Gal}(\mathbb{Q}(i)/\mathbb{Q})$ consists of the identity automorphism and $\sigma$. So, $\text{Gal}(\mathbb{Q}(i)/\mathbb{Q}) \cong C_2$, the cyclic group of order 2.

The Fundamental Theorem of Galois Theory

This theorem establishes a one-to-one correspondence between the intermediate fields of a Galois extension $K/F$ and the subgroups of its Galois group $\text{Gal}(K/F)$.

Let $K/F$ be a Galois extension with Galois group $G = \text{Gal}(K/F)$.

• For every intermediate field $E$ ($F \subseteq E \subseteq K$), there is a unique subgroup $H \subseteq G$ such that $E = K^H$, the fixed field of $H$. The fixed field $K^H$ is defined as $\{x \in K \mid \sigma(x) = x \text{ for all } \sigma \in H\}$.
• For every subgroup $H \subseteq G$, there is a unique intermediate field $E$ ($F \subseteq E \subseteq K$) such that $E = K^H$.

This correspondence has crucial properties:

• If $E_1 \subseteq E_2$ are intermediate fields, then their corresponding subgroups $H_1, H_2$ satisfy $H_2 \subseteq H_1$.
• The degree of the extension $[K:E]$ is equal to the order of the corresponding subgroup $|H|$.
• The degree of the extension $[E:F]$ is equal to the index of the subgroup $|G:H|$.
• An intermediate field $E$ is a Galois extension of $F$ if and only if its corresponding subgroup $H$ is a normal subgroup of $G$. In this case, $\text{Gal}(E/F) \cong G/H$.

This theorem allows us to study the structure of field extensions by studying the structure of subgroups of the Galois group, and vice versa.

Solvability by Radicals

A polynomial equation $p(x) = 0$ is solvable by radicals if its roots can be expressed using only the coefficients of the polynomial, the basic arithmetic operations (addition, subtraction, multiplication, division), and the extraction of $n$-th roots (radicals).

Galois theory provides a criterion for solvability by radicals: A polynomial $p(x) \in F[x]$ is solvable by radicals if and only if its Galois group over $F$ is a solvable group.

A group $G$ is solvable if it has a subnormal series of subgroups $1 = G_0 \triangleleft G_1 \triangleleft \dots \triangleleft G_n = G$ such that each factor group $G_{i+1}/G_i$ is abelian.

For example, the symmetric group $S_n$ is solvable if and only if $n \le 4$. This explains why polynomial equations of degree 1, 2, 3, and 4 have general formulas for their roots involving radicals (e.g., the quadratic formula), but there is no general formula for the roots of a quintic (degree 5) or higher degree polynomial equation using radicals. The Galois group of a generic polynomial of degree $n \ge 5$ is the symmetric group $S_n$, which is not solvable for $n \ge 5$.

Finite Fields

A finite field is a field that contains a finite number of elements. They are also known as Galois fields. Finite fields play a crucial role in number theory, cryptography, coding theory, and algebraic geometry.

Let $F$ be a finite field. There exists a unique prime number $p$ such that $p \cdot 1 = 0$ in $F$, where $1$ is the multiplicative identity of $F$. This prime $p$ is called the characteristic of the field. The smallest field contained within $F$ is the prime field, which is isomorphic to $\mathbb{Z}_p$, the field of integers modulo $p$.

If a finite field $F$ has characteristic $p$, then $F$ can be viewed as a vector space over its prime field $\mathbb{Z}_p$. If $F$ has $q$ elements, then $q$ must be a power of $p$. That is, $|F| = p^n$ for some positive integer $n$.

Example: $\mathbb{Z}_2 = \{0, 1\}$ is a finite field of characteristic 2. $\mathbb{Z}_3 = \{0, 1, 2\}$ is a finite field of characteristic 3.

Construction of Finite Fields

A finite field with $p^n$ elements, where $p$ is a prime and $n \ge 1$, can be constructed as the splitting field of the polynomial $x^{p^n} - x$ over the prime field $\mathbb{Z}_p$.

Let $F_{p^n}$ denote the finite field with $p^n$ elements. The elements of $F_{p^n}$ are precisely the roots of the polynomial $x^{p^n} - x$.

Construction Steps:

1. Choose a prime $p$ and a positive integer $n$.
2. Consider the polynomial $f(x) = x^{p^n} - x$ over the field $\mathbb{Z}_p$.
3. Find an irreducible polynomial $g(x)$ of degree $n$ over $\mathbb{Z}_p$. Such a polynomial always exists.
4. Construct the field extension $\mathbb{Z}_p[x] / \langle g(x) \rangle$. This field is isomorphic to $F_{p^n}$.

Example: Constructing $F_4$. Here $p=2$ and $n=2$, so $p^n = 4$. We need to find an irreducible polynomial of degree 2 over $\mathbb{Z}_2$. The polynomials of degree 2 over $\mathbb{Z}_2$ are $x^2$, $x^2+1$, $x^2+x$, $x^2+x+1$.

• $x^2$ is reducible ($x \cdot x$).
• $x^2+1 = (x+1)^2$ over $\mathbb{Z}_2$, so it's reducible.
• $x^2+x = x(x+1)$ over $\mathbb{Z}_2$, so it's reducible.
• $x^2+x+1$ has no roots in $\mathbb{Z}_2$ (0 gives 1, 1 gives 1+1+1 = 1), so it is irreducible over $\mathbb{Z}_2$.

Thus, $F_4 \cong \mathbb{Z}_2[x] / \langle x^2+x+1 \rangle$. The elements are of the form $a + bx$, where $a, b \in \mathbb{Z}_2$, and $x^2+x+1 = 0$ (so $x^2 = x+1$). The elements are $0, 1, x, x+1$.

The four elements are:

• $0$ (additive identity)
• $1$ (multiplicative identity)
• $x$
• $x+1$

Addition is modulo 2. For example, $x + (x+1) = 2x + 1 = 1$. Multiplication: $x \cdot x = x^2 = x+1$. $x \cdot (x+1) = x^2 + x = (x+1) + x = 2x+1 = 1$.

Properties of Finite Fields

Let $F_{p^n}$ be the finite field with $p^n$ elements.

• Frobenius Automorphism: For any finite field $F_q$ with $q=p^n$ elements, the map $\phi: F_q \to F_q$ defined by $\phi(x) = x^p$ is an automorphism called the Frobenius automorphism. It fixes the prime subfield $\mathbb{Z}_p$.
• Galois Group: The Galois group of $F_{p^n}$ over $\mathbb{Z}_p$ is cyclic of order $n$, generated by the Frobenius automorphism $\phi(x) = x^p$. That is, $\text{Gal}(F_{p^n}/\mathbb{Z}_p) = \langle \phi \rangle$, and $|\text{Gal}(F_{p^n}/\mathbb{Z}_p)| = n$.
• Subfields: The subfields of $F_{p^n}$ are in one-to-one correspondence with the divisors of $n$. If $d$ is a divisor of $n$, there is a unique subfield of $F_{p^n}$ with $p^d$ elements. This subfield is $F_{p^d}$, and it is the fixed field of the subgroup $\langle \phi^{d} \rangle$ of the Galois group.
• Uniqueness: For any prime $p$ and any positive integer $n$, there exists a unique finite field (up to isomorphism) with $p^n$ elements. This field is denoted $F_{p^n}$ or $\text{GF}(p^n)$.

The Multiplicative Group of a Finite Field

Let $F_q$ be a finite field with $q$ elements. The set of non-zero elements, $F_q^* = F_q \setminus \{0\}$, forms a multiplicative group under the field multiplication.

Theorem: The multiplicative group $F_q^*$ of a finite field $F_q$ is cyclic and has order $q-1$.

This means there exists an element $\alpha \in F_q^*$ such that every non-zero element of $F_q$ can be written as a power of $\alpha$. Such an element $\alpha$ is called a primitive element of $F_q$.

Example: In $F_4 = \{0, 1, x, x+1\}$, the multiplicative group is $F_4^* = \{1, x, x+1\}$. The order is $4-1=3$. The element $x$ is a generator:

• $x^1 = x$
• $x^2 = x+1$
• $x^3 = x \cdot x^2 = x(x+1) = 1$ (since $x^2+x = (x+1)+x = 1$ in $F_4$)

So, $F_4^*$ is cyclic, generated by $x$.

Applications of Finite Fields

Finite fields are fundamental in several areas of mathematics and computer science:

• Cryptography: Elliptic curve cryptography (ECC) and other public-key cryptosystems rely heavily on computations over finite fields, particularly $F_{p^n}$ and binary fields $F_{2^n}$.
• Coding Theory: Error-correcting codes, such as Reed-Solomon codes used in CDs, DVDs, and QR codes, are constructed using the algebraic properties of finite fields.
• Experimental Design: In statistics and design of experiments, finite fields are used to construct balanced incomplete block designs (BIBDs).
• Computer Science: Hashing algorithms, pseudo-random number generators, and computational algebra systems often utilize finite field arithmetic.

Galois Theory and Solvability of Polynomials (Summary)

Galois theory provides a profound link between the roots of polynomials and group theory.

Key connections:

• The symmetries of the roots of a polynomial are captured by its Galois group.
• The structure of the Galois group determines whether the roots can be expressed using radicals.
• A polynomial is solvable by radicals if and only if its Galois group is a solvable group.
• The lack of a general formula for the roots of quintic and higher-degree polynomials is directly explained by the non-solvability of the symmetric groups $S_n$ for $n \ge 5$.

Finite Fields and their Galois Groups (Summary)

Finite fields $F_{p^n}$ offer a rich structure for algebraic study.

Key connections:

• Finite fields exist if and only if their order is $p^n$ for some prime $p$ and integer $n \ge 1$.
• The field $F_{p^n}$ is the splitting field of $x^{p^n} - x$ over $\mathbb{Z}_p$.
• The Galois group $\text{Gal}(F_{p^n}/\mathbb{Z}_p)$ is cyclic of order $n$, generated by the Frobenius automorphism $x \mapsto x^p$.
• Subfields of $F_{p^n}$ correspond to subgroups of its Galois group, which in turn correspond to divisors of $n$.
• The multiplicative group $F_q^*$ is always cyclic of order $q-1$.

Understanding Galois theory and finite fields is essential for advanced topics in algebra, number theory, and their applications.