Field (Mathematics) — Definition, Formula & Examples

A field is a set of elements equipped with two operations, addition and multiplication, where you can add, subtract, multiply, and divide (except by zero) and always stay within the set. Familiar examples include the rational numbers

\mathbb{Q}

, the real numbers

\mathbb{R}

, and the complex numbers

\mathbb{C}

A field is a set $F$ together with two binary operations $+$ and $\cdot$ such that $(F, +)$ is an abelian group with identity $0$ , $(F \setminus \{0\}, \cdot)$ is an abelian group with identity $1$ , and multiplication distributes over addition: $a \cdot (b + c) = a \cdot b + a \cdot c$ for all $a, b, c \in F$ .

How It Works

To verify that a set with two operations forms a field, you check that both operations are associative and commutative, that additive and multiplicative identities exist, that every element has an additive inverse, and that every nonzero element has a multiplicative inverse. You also confirm the distributive law holds. If any one of these axioms fails, the structure is not a field. For instance, the integers

\mathbb{Z}

satisfy every axiom except the existence of multiplicative inverses (e.g., there is no integer

n

with

2n = 1

), so

\mathbb{Z}

is a ring but not a field.

Worked Example

Problem: Show that the set

\mathbb{Z}_5 = \{0, 1, 2, 3, 4\}

with addition and multiplication modulo 5 is a field.

Step 1: Check that addition mod 5 forms an abelian group. The identity is 0, and every element has an additive inverse: for example,

3 + 2 \equiv 0 \pmod{5}

, so the inverse of 3 is 2. Addition mod 5 is associative and commutative.

Step 2: Check that the nonzero elements form an abelian group under multiplication mod 5. The identity is 1. Each nonzero element has a multiplicative inverse:

1 \cdot 1 = 1,\quad 2 \cdot 3 = 6 \equiv 1,\quad 4 \cdot 4 = 16 \equiv 1 \pmod{5}

Step 3: Verify the distributive law. For example,

2 \cdot (3 + 4) = 2 \cdot 2 = 4

and

2 \cdot 3 + 2 \cdot 4 = 6 + 8 = 14 \equiv 4 \pmod{5}

. Both sides agree.

Answer: All field axioms are satisfied, so

\mathbb{Z}_5

is a field with 5 elements.

Why It Matters

Fields underpin linear algebra (vector spaces are defined over fields), number theory (algebraic number fields), and cryptography (finite fields like

\mathbb{Z}_p

secure modern encryption). Understanding field axioms is essential in any abstract algebra or advanced mathematics course.

Common Mistakes

Mistake: Assuming the integers

\mathbb{Z}

form a field because they satisfy most of the axioms.

Correction: The integers lack multiplicative inverses for elements other than

\pm 1

. For example,

\frac{1}{2} \notin \mathbb{Z}

. The integers form a ring, not a field.