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examples of fields
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(Example)
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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.
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Cross-references: absolute value, canonical, total order, topological field, topology, structure, field of fractions, integral domain, expressions, variable, formal Laurent series, information, residue field, point, rational functions, scheme, variety, Riemann surface, functions, meromorphic, open set, connected, polynomial functions, function field, coefficients, polynomials, quotients, field of rational functions, power, elements, finite field, integers, valuation, completion, prime number, simple, finite field extension, separable, algebraic number fields, countable, computable numbers, formula, definable, Turing machine, sequence, digit, computable, algebraic closure, algebraic numbers, proper class, strict, infinitesimal, surreal numbers, numbers, hyperreal, complex numbers, real numbers, rational numbers, division, multiplication, subtraction, addition, operations
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This is version 13 of examples of fields, born on 2002-07-09, modified 2004-06-01.
Object id is 3162, canonical name is ExamplesOfFields.
Accessed 9750 times total.
Classification:
| AMS MSC: | 12E99 (Field theory and polynomials :: General field theory :: Miscellaneous) |
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Pending Errata and Addenda
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