3.8 Fields

Definition of fields

A field is an algebraic structure consisting of a set equipped with two operations (addition and multiplication) that satisfy a specific list of axioms. Fields give you a system where you can add, subtract, multiply, and divide (except by zero) and have everything behave the way you'd expect from working with rational or real numbers.

Why care about fields? They're the structure that makes equation-solving work. Whenever you manipulate an equation by "dividing both sides by $x$" or "subtracting 3 from both sides," you're relying on field axioms whether you realize it or not.

Axioms of fields

A set $F$ with operations $+$ and $\cdot$ is a field if and only if all of the following hold for every $a, b, c \in F$:

• Closure: $a + b \in F$ and $a \cdot b \in F$
• Commutativity: $a + b = b + a$ and $a \cdot b = b \cdot a$
• Associativity: $(a + b) + c = a + (b + c)$ and $(a \cdot b) \cdot c = a \cdot (b \cdot c)$
• Distributivity: $a \cdot (b + c) = a \cdot b + a \cdot c$
• Additive identity: There exists an element $0 \in F$ such that $a + 0 = a$
• Multiplicative identity: There exists an element $1 \in F$ (with $1 \neq 0$) such that $a \cdot 1 = a$
• Additive inverses: For every $a$, there exists $-a \in F$ with $a + (-a) = 0$
• Multiplicative inverses: For every $a \neq 0$, there exists $a^{-1} \in F$ with $a \cdot a^{-1} = 1$

The requirement that $1 \neq 0$ means the trivial set $\{0\}$ doesn't count as a field. Every field has at least two elements.

Examples of fields

• Rational numbers $\mathbb{Q}$: The most basic infinite field. Every fraction $p/q$ (with $q \neq 0$) has an inverse, and all axioms hold under standard arithmetic.
• Real numbers $\mathbb{R}$: Extends $\mathbb{Q}$ by including irrationals like $\sqrt{2}$ and $\pi$. This is the field underlying calculus.
• Complex numbers $\mathbb{C}$: Extends $\mathbb{R}$ by including $i = \sqrt{-1}$, which guarantees every polynomial equation has a root.
• Finite fields (Galois fields): The smallest example is $GF(2) = \{0, 1\}$ with arithmetic mod 2. These are heavily used in cryptography and coding theory.

Non-examples of fields

Understanding why something fails to be a field is just as useful as knowing the examples.

• Integers $\mathbb{Z}$: The number 2 has no multiplicative inverse in $\mathbb{Z}$ (there's no integer $n$ with $2n = 1$). So $\mathbb{Z}$ is a ring, not a field.
• Matrices under standard operations: Matrix multiplication is not commutative ($AB \neq BA$ in general), violating the commutativity axiom. Also, not every nonzero matrix is invertible.
• Polynomials with real coefficients: The polynomial $x$ has no multiplicative inverse that's also a polynomial. (The inverse would be $1/x$, which isn't a polynomial.)
• Quaternions $\mathbb{H}$: These satisfy every field axiom except commutativity of multiplication. Quaternions form what's called a division ring (or skew field).

Properties of fields

The axioms above interact to produce several useful consequences. You don't need to memorize these as separate axioms; they all follow from the ones listed above.

Commutativity in fields

Both operations commute: $a + b = b + a$ and $a \cdot b = b \cdot a$ for all elements. This is what separates fields from division rings.

Commutativity gives you the cancellation law: if $a \cdot b = a \cdot c$ and $a \neq 0$, you can conclude $b = c$. The proof is straightforward: multiply both sides by $a^{-1}$.

Associativity in fields

Associativity of both addition and multiplication means you can write expressions like $a + b + c$ or $a \cdot b \cdot c$ without ambiguity. Parentheses don't matter for chains of the same operation.

This property is what lets you define powers like $a^n = a \cdot a \cdot \ldots \cdot a$ without worrying about grouping.

Distributivity in fields

The distributive law $a \cdot (b + c) = a \cdot b + a \cdot c$ is the bridge between addition and multiplication. Right distributivity $(a + b) \cdot c = a \cdot c + b \cdot c$ also holds, and in a commutative setting it follows directly from left distributivity plus commutativity.

Distributivity is what makes factoring and expanding expressions possible. It's also the key to proving that $a \cdot 0 = 0$ for any field element $a$.

Identity elements

• The additive identity $0$ satisfies $a + 0 = a$ for all $a$.
• The multiplicative identity $1$ satisfies $a \cdot 1 = a$ for all $a$.

Both identities are unique. If you suppose there were two additive identities $0$ and $0'$, then $0 = 0 + 0' = 0'$, so they must be the same element. The same argument works for $1$.

Field operations

Addition in fields

Addition in a field is closed, commutative, and associative. Every element $a$ has an additive inverse $-a$ satisfying $a + (-a) = 0$. Subtraction is then defined as $a - b = a + (-b)$.

Multiplication in fields

Multiplication is closed, commutative, and associative, and distributes over addition. The multiplicative identity $1$ is the neutral element. Multiplication by $0$ always yields $0$.

A critical property of fields: there are no zero divisors. If $a \cdot b = 0$, then either $a = 0$ or $b = 0$. This follows from the existence of multiplicative inverses.

Inverse elements

• Additive inverse: Every element $a$ has a unique inverse $-a$ with $a + (-a) = 0$.
• Multiplicative inverse: Every nonzero element $a$ has a unique inverse $a^{-1}$ with $a \cdot a^{-1} = 1$.

Division is defined using multiplicative inverses: $a / b = a \cdot b^{-1}$ for $b \neq 0$. This is why division by zero is undefined: $0$ has no multiplicative inverse.

To see why inverses are unique, suppose $a$ has two multiplicative inverses $b$ and $c$. Then $b = b \cdot 1 = b \cdot (a \cdot c) = (b \cdot a) \cdot c = 1 \cdot c = c$.

Subfields and extensions

Subfield definition

A subfield of a field $F$ is a subset $K \subseteq F$ that is itself a field under the same operations. To verify that $K$ is a subfield, check that:

1. $K$ is closed under addition, subtraction, and multiplication
2. $K$ is closed under taking multiplicative inverses (for nonzero elements)
3. $K$ contains $0$ and $1$

Every field has a prime subfield, which is the smallest subfield it contains. For fields of characteristic 0, the prime subfield is (isomorphic to) $\mathbb{Q}$. For fields of characteristic $p$, it's $\mathbb{F}_p$.

Field extensions

A field extension $L/K$ is a pair where $L$ is a field containing $K$ as a subfield. You can think of $L$ as a "bigger" field built on top of $K$.

The degree of the extension, written $[L:K]$, is the dimension of $L$ when viewed as a vector space over $K$. For example, $[\mathbb{C}:\mathbb{R}] = 2$ because every complex number can be written as $a + bi$ using two real-number coordinates.

A simple extension is one generated by adding a single element to the base field. For instance, $\mathbb{Q}(\sqrt{2}) = \{a + b\sqrt{2} \mid a, b \in \mathbb{Q}\}$ is a simple extension of $\mathbb{Q}$ with degree 2.

Algebraic vs transcendental extensions

• An element $\alpha \in L$ is algebraic over $K$ if it satisfies some polynomial equation with coefficients in $K$. For example, $\sqrt{2}$ is algebraic over $\mathbb{Q}$ because it satisfies $x^2 - 2 = 0$.
• An element is transcendental over $K$ if no such polynomial exists. Both $\pi$ and $e$ are transcendental over $\mathbb{Q}$.

An extension is called algebraic if every element of $L$ is algebraic over $K$, and transcendental otherwise. Algebraic extensions can have finite or infinite degree, but simple algebraic extensions always have finite degree. Transcendental extensions always have infinite degree.

Characteristic of fields

The characteristic of a field is the smallest positive integer $p$ such that adding $1$ to itself $p$ times gives $0$. If no such integer exists, the characteristic is 0.

Zero vs prime characteristic

• Characteristic 0: Adding $1 + 1 + 1 + \ldots$ never reaches $0$. Fields like $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ all have characteristic 0. These fields contain a copy of $\mathbb{Q}$ as their prime subfield.
• Characteristic $p$ (where $p$ is prime): $\underbrace{1 + 1 + \cdots + 1}_{p \text{ times}} = 0$. The characteristic of a field is always either 0 or a prime number. (If it were composite, say $p = ab$, then the field would have zero divisors, which is impossible.) These fields contain a copy of $\mathbb{F}_p$ as their prime subfield.

Finite vs infinite fields

• Finite fields always have prime characteristic. A finite field has exactly $p^n$ elements, where $p$ is its characteristic and $n$ is a positive integer. For each prime power $p^n$, there is exactly one finite field (up to isomorphism), denoted $\mathbb{F}_{p^n}$ or $GF(p^n)$.
• Infinite fields can have characteristic 0 (like $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$) or prime characteristic (like the field of rational functions $\mathbb{F}_p(x)$).

Field isomorphisms

Definition of isomorphism

A field isomorphism is a bijection $f: F \to G$ between two fields that preserves both operations:

• $f(a + b) = f(a) + f(b)$
• $f(a \cdot b) = f(a) \cdot f(b)$

These two conditions automatically guarantee that $f(0_F) = 0_G$ and $f(1_F) = 1_G$. (You don't need to state those separately.) If such a function exists, the fields $F$ and $G$ are isomorphic, written $F \cong G$, meaning they have identical algebraic structure even if their elements look different.

Properties of isomorphisms

• Isomorphic fields share all algebraic properties: same characteristic, same cardinality, same number of solutions to any polynomial equation.
• The composition of two isomorphisms is an isomorphism.
• The inverse of an isomorphism is also an isomorphism.
• The identity map is always an isomorphism from a field to itself.

Examples of isomorphic fields

• $\mathbb{C}$ is isomorphic to the field of ordered pairs $(a, b) \in \mathbb{R}^2$ with the operations $(a,b) + (c,d) = (a+c, b+d)$ and $(a,b) \cdot (c,d) = (ac - bd, ad + bc)$. This is just a formal way of writing $a + bi$.
• Any two finite fields of the same order are isomorphic. For instance, any field with 4 elements is isomorphic to $\mathbb{F}_2[x]/(x^2 + x + 1)$.
• The algebraic closure of $\mathbb{Q}$ (the set of all algebraic numbers) is isomorphic to the field of algebraic numbers sitting inside $\mathbb{C}$.

Applications of fields

Fields in algebra

Fields provide the setting for solving polynomial equations and studying their roots. Linear algebra is built on fields: a vector space is defined over a field, and the scalars come from that field. Galois theory uses field extensions to determine which polynomial equations can be solved by radicals, famously showing that the general quintic cannot.

Fields in number theory

Algebraic number theory studies number fields, which are finite extensions of $\mathbb{Q}$. These are used to analyze Diophantine equations and properties of prime numbers in extended number systems. Finite fields are essential in modern cryptography: protocols like RSA and elliptic curve cryptography rely on arithmetic in $\mathbb{F}_p$ and its extensions.

Fields in geometry

Analytic geometry assigns coordinates from a field to geometric points. The choice of field matters: working over $\mathbb{R}$ gives classical geometry, while working over finite fields gives finite geometries used in combinatorics. Algebraic geometry studies solution sets of polynomial equations over various fields, connecting algebra and geometry in deep ways.

Special types of fields

Ordered fields

An ordered field is a field with a total order $<$ that's compatible with the field operations: if $a < b$, then $a + c < b + c$, and if $a > 0$ and $b > 0$, then $a \cdot b > 0$.

$\mathbb{Q}$ and $\mathbb{R}$ are ordered fields. $\mathbb{C}$ is not an ordered field. (There's no way to define $<$ on $\mathbb{C}$ that's compatible with both operations, since you'd need $i^2 = -1 > 0$ or $i^2 = -1 < 0$, and both lead to contradictions.)

Complete fields

A field is complete if every Cauchy sequence in it converges to an element of the field. $\mathbb{R}$ is the unique complete ordered field (up to isomorphism). $\mathbb{Q}$ is not complete: the sequence $1, 1.4, 1.41, 1.414, \ldots$ converges to $\sqrt{2}$, which isn't in $\mathbb{Q}$.

Completeness is what makes calculus work. Without it, you can't guarantee that limits, derivatives, and integrals exist.

Algebraically closed fields

A field is algebraically closed if every non-constant polynomial with coefficients in the field has at least one root in the field. $\mathbb{C}$ is algebraically closed (this is the Fundamental Theorem of Algebra). $\mathbb{R}$ is not: $x^2 + 1 = 0$ has no real solution.

Every field has an algebraic closure, which is the smallest algebraically closed field containing it. The algebraic closure of $\mathbb{Q}$ is the field of algebraic numbers $\overline{\mathbb{Q}}$, which is a proper subset of $\mathbb{C}$.

Field theory

Fundamental theorem of algebra

Every non-constant polynomial with coefficients in $\mathbb{C}$ has at least one root in $\mathbb{C}$. As a consequence, every polynomial of degree $n$ over $\mathbb{C}$ factors completely into $n$ linear factors (counting multiplicity).

Despite its name, most proofs of this theorem use analysis or topology rather than pure algebra. Common proof strategies include arguments from complex analysis (Liouville's theorem), topology (winding numbers), or a mix of algebra and analysis.

Galois theory basics

Galois theory connects field extensions to group theory. Given a polynomial $f(x)$ over a field $K$, you build its splitting field $L$ (the smallest extension where $f$ factors completely). The Galois group $\text{Gal}(L/K)$ consists of all field automorphisms of $L$ that fix every element of $K$.

The Fundamental Theorem of Galois Theory establishes a correspondence between intermediate fields (fields between $K$ and $L$) and subgroups of the Galois group. This correspondence is what lets you translate questions about polynomial solvability into questions about group structure. The classical result: a polynomial is solvable by radicals if and only if its Galois group is a solvable group.

Field automorphisms

A field automorphism is an isomorphism from a field to itself. The set of all automorphisms of a field forms a group under composition.

Key examples:

• Complex conjugation ($a + bi \mapsto a - bi$) is an automorphism of $\mathbb{C}$ that fixes $\mathbb{R}$.
• The Frobenius automorphism $\phi(x) = x^p$ is an automorphism of any finite field of characteristic $p$. It generates the Galois group of finite field extensions.
• $\mathbb{Q}$ has only the identity automorphism. (Every automorphism must fix $0$ and $1$, and from there it must fix every rational number.)

Computational aspects

Field arithmetic

How you actually compute in a field depends on which field you're working in. In $\mathbb{Q}$, you do fraction arithmetic. In finite fields $\mathbb{F}_p$, you do modular arithmetic (all operations mod $p$). In extension fields like $\mathbb{F}_{p^n}$, you work with polynomials modulo an irreducible polynomial of degree $n$.

Algorithms for field operations

• Fast multiplication: Karatsuba's algorithm multiplies large numbers faster than the grade-school method ($O(n^{1.585})$ vs $O(n^2)$ for $n$-digit numbers). For very large inputs, FFT-based methods are even faster.
• Modular exponentiation: The square-and-multiply algorithm computes $a^k \mod n$ in $O(\log k)$ multiplications, which is essential for cryptographic applications.
• Polynomial factorization over finite fields: Berlekamp's algorithm and the Cantor-Zassenhaus algorithm factor polynomials over $\mathbb{F}_p$ efficiently, which matters for constructing and working with finite field extensions.

Complexity considerations

Field operations have different computational costs depending on the representation. Addition and subtraction in $\mathbb{F}_p$ are cheap (linear in the number of digits of $p$), while multiplication is more expensive, and computing inverses (via the extended Euclidean algorithm) costs more still. For large-scale applications like cryptography, choosing efficient representations and algorithms for field arithmetic directly impacts performance.