examples of fields

Fields (http://planetmath.org/Field) are typically sets of “numbers” in which the arithmeticoperations of addition, subtraction, multiplication and division aredefined. The following is a list of examples of fields.

• The set of all rational numbers $\mathbb{Q}$, all real numbers $ℝ$ and allcomplex numbers $ℂ$ are the most familiar examples of fields.

• Slightly more exotic, the hyperreal numbers and the surrealnumbers are fields containing infinitesimal and infinitely largenumbers. (The surreal numbers aren’t a field in the strict sense sincethey form a proper class and not a set.)

• The algebraic numbers form a field; this is the algebraicclosure of $\mathbb{Q}$. In general, every field has an (essentiallyunique) 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 beprecisely specified using a logical formula) form a field containing the computable numbers; arguably, thisfield 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 $\mathbb{Q}$ by adjoining some (finite number of) algebraic numbers. For instance $\mathbb{Q}(\sqrt{2})=\{u+v\sqrt{2}\mid u,v\in \mathbb{Q}\}$ and $\mathbb{Q}(\sqrt[3]{2},i)=\{u+vi+w\sqrt[3]{2}+xi\sqrt[3]{2}+y\sqrt[3]{4}+zi\sqrt[3]{4}\mid u,v,w,x,y,z\in \mathbb{Q}\}=\mathbb{Q}(i\sqrt[3]{2})$ (every separable finite field extension is simple).

• If $p$ is a prime number, then the $p$-adic numbers form afield ${\mathbb{Q}}_p$ 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 afinite field with $p$ elements, typically denoted by ${𝔽}_p$. Moregenerally, for every prime (http://planetmath.org/Prime) power $p^n$ there is one and only onefinite field ${𝔽}_{p^n}$ with $p^n$ elements.

• If $K$ is a field, we can form the field of rational functionsover $K$, denoted by $K(X)$. It consists of quotients of polynomialsin $X$ with coefficients in $K$.

• If $V$ is a variety (http://planetmath.org/AffineVariety) 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 $ℂ$, then theset 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 thevariable $X$ consistsof all expressions of the form

  $$\sum _{j=-M}^{\mathrm{\infty }}{a}_j{X}^j$$

  where $M$ is some integer and the coefficients $a_j$ come from $K$.

• More generally, whenever $R$ is an integral domain, we can formits field of fractions, a field whose elements are thefractions 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.