Field¶

Definition¶

A field is an algebraic structure $(F,+,\cdot)$ in which addition and multiplication behave like ordinary arithmetic: addition makes $F$ an Abelian group, multiplication is associative and commutative and has an identity $1\neq0$$0$, every nonzero element has a multiplicative inverse, and multiplication distributes over addition.

When a vector space is defined over a field $F$, the elements of $F$ are called scalars.

Interpretation¶

A field is a number system in which the four basic operations work as for the real numbers: one can add, subtract, multiply, and divide by any nonzero element. Equivalently, a field is a commutative ring with an identity $1\ne0$ in which every nonzero element has a multiplicative inverse.

Each part of the definition supplies one operation. The Abelian group $(F,+)$ provides $0$ and negatives $-a$, hence subtraction $a-b=a+(-b)$$(-b)$. The multiplicative inverses provide division, $a/b=a\cdot b^{-1}$ for $b\neq0$$0$. Distributivity is the only rule that links the two operations, and familiar facts such as $0\cdot a=0$ and $(-1)\cdot a=-a$ follow from it.

$\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$ are fields; $\mathbb{Z}$ is not, since $2x=1$ has no integer solution. Fields can also be finite. The set $\{0,1\}$, with sums and products replaced by their remainders after division by $2$, is a field. The same construction with remainders after division by $4$ is not, because $2\cdot x$ leaves the remainder $0$ or $2$ for every integer $x$, never $1$.

A common slip is to read the inverse condition as applying to all elements. It is required only for nonzero elements, and $0$ never has a multiplicative inverse, because $0\cdot b=0\neq1$ for every $b$. So division by zero is undefined in every field.

Fields supply the scalars of vector spaces, most often $\mathbb{R}$ or $\mathbb{C}$; $\mathbb{C}$ has the advantage of being algebraically closed.

Key Results¶

• In a vector space over $F$, scalar multiplication uses elements of $F$.
• The real numbers and complex numbers are fields; the integers are not a field because most nonzero integers do not have integer multiplicative inverses.

Examples¶

• $\mathbb{R}$ with ordinary addition and multiplication is a field.
• $\mathbb{C}$ with ordinary addition and multiplication is a field.

Prerequisites¶

• Ring
• Abelian group

Related Concepts¶

• Real Number
• Complex Number
• Algebraic Structure
• Commutativity
• Complex Addition
• Inverse Element
• Distributivity

Sources¶

• Math II: Linear Algebra · Weeks 1–2

Aliases: field of scalars, base field Keywords: scalar, scalars Symbols: complex numbers, integers, rational numbers, real numbers

Last updated: 20 October 2026 03:02