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In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.
Fields are important objects of study in algebra since they provide a useful generalization of many number systems, such as the rational numbers, real numbers, and complex numbers.
For a given field F, the set F(X) of rational functions in the variable X with coefficients in F is a field; this is defined as the set of quotients of polynomials with coefficients in F.
In mathematics, a local field is a special type of field which has the additional property that it is a complete metric space with respect to a discrete valuation.
There is some inconsistency in usage, but usually a local field is further assumed to be locally compact, and often the field of real numbers and the field of complex numbers are considered to be local as well by virtue of their local compactness.
A local field of characteristic p can always be realized as the field of Laurent series in one variable with coefficients in a finite field (also of characteristic p).
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