Fields are typically sets of ``numbers'' in which the arithmetic operations of addition, subtraction, multiplication and division are defined. The following is a list of examples of fields.

• The set of all rational numbers $\Bbb{Q}$ , all real numbers $\Bbb{R}$ and all complex numbers $\Bbb{C}$ are the most familiar examples of fields.
• Slightly more exotic, the hyperreal numbers and the surreal numbers are fields containing infinitesimal and infinitely large numbers. (The surreal numbers aren't a field in the strict sense since they form a proper class and not a set.)
• The algebraic numbers form a field; this is the algebraic closure of $\Bbb{Q}$ . In general, every field has an (essentially unique) algebraic closure.
• The computable complex numbers (those whose digit sequence can be produced by a Turing machine) form a field. The definable complex numbers (those which can be precisely specified using a logical formula) form a field containing the computable numbers; arguably, this field contains all the numbers we can ever talk about. It is countable.
• The so-called algebraic number fields (sometimes just called number fields) arise from $\Bbb{Q}$ by adjoining some (finite number of) algebraic numbers. For instance $\Bbb{Q}(\sqrt{2}) = \{u + v\sqrt{2} \mid u,v\in\Bbb{Q}\}$ and $\Bbb{Q}(\sqrt[3]{2},i) = \{u + vi + w\sqrt[3]{2} + xi\sqrt[3]{2} + y \sqrt[3]{4} + zi \sqrt[3]{4} \mid u,v,w,x,y,z\in\Bbb{Q}\} = \mathbb{Q}(i\sqrt[3]{2})$ (every separable finite field extension is simple).
• If $p$ is a prime number, then the $p$ -adic numbers form a field $\Bbb{Q}_p$ which is the completion of the field $\mathbb{Q}$ with respect to the $p$ -adic valuation.
• If $p$ is a prime number, then the integers modulo $p$ form a finite field with $p$ elements, typically denoted by $\Bbb{F}_p$ . More generally, for every prime power $p^n$ there is one and only one finite field $\Bbb{F}_{p^n}$ with $p^n$ elements.
• If $K$ is a field, we can form the field of rational functions over $K$ , denoted by $K(X)$ . It consists of quotients of polynomials in $X$ with coefficients in $K$ .
• If $V$ is a variety over the field $K$ , then the function field of $V$ , denoted by $K(V)$ , consists of all quotients of polynomial functions defined on $V$ .
• If $U$ is a domain (= connected open set) in $\Bbb{C}$ , then the set of all meromorphic functions on $U$ is a field. More generally, the meromorphic functions on any Riemann surface form a field.
• If $X$ is a variety (or scheme) then the rational functions on $X$ form a field. At each point of $X$ , there is also a residue field which contains information about that point.
• The field of formal Laurent series over the field $K$ in the variable $X$ consists of all expressions of the form $$\sum_{j=-M}^\infty a_j X^j$$ where $M$ is some integer and the coefficients $a_j$ come from $K$ .
• More generally, whenever $R$ is an integral domain, we can form its field of fractions, a field whose elements are the fractions of elements of $R$ .

Many of the fields described above have some sort of additional structure, for example a topology (yielding a topological field), a total order, or a canonical absolute value.