# quotient field

## quotient field

[′kwō·shənt ‚fēld]
(mathematics)
The smallest field containing a given integral domain; obtained by formally introducing all quotients of elements of the integral domain.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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The measurements are based on what is called an Environmental Impact Quotient Field Use rating - a method of measuring the impacts of herbicide use based on toxicity, leaching, runoff, soil half-life and other factors of 100 pesticides approved for use by the U.S.
We recall that an extension R [subset or equal to] S of a normal domain R of dimension two is called Galois if S is the integral closure of R in L, where K [subset or equal to] L is a finite Galois extension of the quotient field K of R, and R [subset or equal to] S is unramified at all prime ideals of height one.
In the discrete-time case the definition of the transfer function is based on a non-commutative twisted polynomial ring, which is a special case of the skew polynomial ring and can be embedded into its quotient field by the Ore condition.
We also briefly sketch the construction of the quotient field of twisted polynomials.
where the entries of f and g are meromorphic functions, which we think of as elements of the quotient field of the ring of analytic functions, and x(t) [member of] [R.sup.n], u(t) [member of] [R.sup.m], and y(t) [member of] [R.sup.p] denote, respectively, the state, the input, and the output of the system.
The non-commutative ring can be embedded into its quotient field (or field of fractions) if the so-called Ore condition is satisfied.
The ring K[[delta]] can, therefore, be embedded into a non-commutative quotient field [17,18] by defining quotients as
Forming the quotient field of the ring modulo the maximal ideal (that is, applying the canonical place map), one obtains a new field.

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