Field (algebra)

Definition · applies to algebraic structures

High school

A field is a set of numbers (or other objects) that can be added, subtracted, multiplied, and divided by every nonzero element, following the same rules as the arithmetic of real numbers.

Undergraduate (abstract algebra)

Consider a set $F$ with two binary operations, addition $+$ and multiplication $\cdot$. The set $F$ is a field if

1. $F$ is a ring with $+$ and $\cdot$,
2. multiplication is commutative,
3. There exists an additive identity $0$ and a multiplicative identity $1$ such that $0 \neq 1$, and
4. every element $a \neq 0$ has a multiplicative inverse $a^{-1}$ such that $a a^{-1} = 1$.

Examples

Examples

• The rational numbers $\mathbb{Q}$, with addition and multiplication High school
  The rational numbers, with addition and multiplication, form a field, because multiplication is commutative and every nonzero fraction $a/b$ has the inverse $b/a$.
• The integers modulo $5$, $\setdef{0, 1, 2, 3, 4}$, with addition and multiplication modulo $5$ Undergraduate
  The integers modulo $5$ form a field, because every nonzero element has a multiplicative inverse: $1 \cdot 1 = 1$, $2 \cdot 3 = 6 = 1$, and $4 \cdot 4 = 16 = 1$ modulo $5$.

Non-examples

Things that may be mistaken for examples.

• The integers $\mathbb{Z}$, with addition and multiplication High school
  The integers, with addition and multiplication, do not form a field, because most integers do not have multiplicative inverses. For example, $2$ has no multiplicative inverse in $\mathbb{Z}$ because $1/2$ is not an integer.
• The nonzero real numbers, with multiplication High school
  The nonzero real numbers are not a field without $0$ because a field requires there is an additive identity.
• The integers modulo $6$, $\setdef{0, 1, 2, 3, 4, 5}$, with addition and multiplication modulo $6$ Undergraduate
  The integers modulo $6$ do not form a field, because $2$ has no multiplicative inverse: $2 k$ modulo $6$ is $0$, $2$, or $4$, and none of these is $1$. Also, $2 \cdot 3 = 0$ modulo $6$, although neither $2$ nor $3$ is $0$.
• The $2 \times 2$ real matrices, with matrix addition, multiplication by real numbers, and matrix multiplication Undergraduate
  The $2 \times 2$ real matrices do not form a field, because matrix multiplication is not commutative, and nonzero singular matrices, such as \(\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\), have no inverse.

Properties

• A field has no zero divisors: if $a b = 0$, then $a = 0$ or $b = 0$.
• The integers modulo $n$ form a field if and only if $n$ is prime.
• The scalars of a vector space come from a field.

Notes

This “field” is unrelated to a vector field or to a scalar field in physics, which are functions that assign a vector or a number to each point of space.

Distinctions involving this term

• Group vs. ring vs. field: Groups, rings, and fields are sets with operations that obey some of the rules of arithmetic. A group has one operation, which is associative, has an identity element, and can be undone. A ring has an addition and a multiplication; its addition forms a commutative group, but its elements need not have multiplicative inverses. A field is a commutative ring in which every nonzero element has a multiplicative inverse, so division by nonzero elements is possible.

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