examples of fields
Fields (http://planetmath.org/Field) 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.

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The set of all rational numbers $\mathbb{Q}$, all real numbers $\mathbb{R}$ and all complex numbers^{} $\u2102$ are the most familiar examples of fields.

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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.)

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The algebraic numbers^{} form a field; this is the algebraic closure^{} of $\mathbb{Q}$. In general, every field has an (essentially unique) algebraic closure.

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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^{}.

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The socalled 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).

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If $p$ is a prime number^{}, then the $p$adic numbers form a field ${\mathbb{Q}}_{p}$ which is the completion of the field $\mathbb{Q}$ with respect to the $p$adic valuation^{}.

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If $p$ is a prime number, then the integers modulo $p$ form a finite field^{} with $p$ elements, typically denoted by ${\mathbb{F}}_{p}$. More generally, for every prime (http://planetmath.org/Prime) power ${p}^{n}$ there is one and only one finite field ${\mathbb{F}}_{{p}^{n}}$ with ${p}^{n}$ elements.

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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 $K$.

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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 $V$.

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If $U$ is a domain (= connected open set) in $\u2102$, then the set of all meromorphic functions on $U$ is a field. More generally, the meromorphic functions on any Riemann surface^{} form a field.

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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.

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The field of formal Laurent series over the field $K$ in the variable $X$ consists of 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 $K$.

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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^{}.
Title  examples of fields 

Canonical name  ExamplesOfFields 
Date of creation  20130322 12:50:13 
Last modified on  20130322 12:50:13 
Owner  AxelBoldt (56) 
Last modified by  AxelBoldt (56) 
Numerical id  16 
Author  AxelBoldt (56) 
Entry type  Example 
Classification  msc 12E99 
Related topic  NumberField 