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The field of fractions of an integral domain is sometimes denoted by or (), and the construction is sometimes also called the fraction field, field of quotients, or quotient field of . All four are in common usage, but are not to be confused with the quotient of a ring by an ideal , which is a quite different concept.
Let A be an integrally closed domain with field of fractions K and let L be a field extension of K. Then x∈L is integral over A if and only if it is algebraic over K and its minimal polynomial over K has coefficients in A. [1] In particular, this means that any element of L integral over A is root of a monic polynomial in A[X] that is ...
An integral domain is said to be integrally closed if it is equal to its integral closure in its field of fractions. An ordered group G is called integrally closed if for all elements a and b of G, if a n ≤ b for all natural numbers n then a ≤ 1.
Informally, a field is a set, along with two operations defined on that set: an addition operation written as a + b, and a multiplication operation written as a ⋅ b, both of which behave similarly as they behave for rational numbers and real numbers, including the existence of an additive inverse −a for all elements a, and of a multiplicative inverse b −1 for every nonzero element b.
For example, given a commutative ring R, the field of fractions of the quotient ring of R by a prime ideal p can be identified with the residue field of the localization of R at p; that is / (/) (all these constructions can be defined by universal properties).
In the case of coefficients in a unique factorization domain R, "rational numbers" must be replaced by "field of fractions of R". This implies that, if R is either a field, the ring of integers, or a unique factorization domain, then every polynomial ring (in one or several indeterminates) over R is a unique factorization domain. Another ...
An extension is algebraic if and only if its transcendence degree is 0; the empty set serves as a transcendence basis here.; The field of rational functions in n variables K(x 1,...,x n) (i.e. the field of fractions of the polynomial ring K[x 1,...,x n]) is a purely transcendental extension with transcendence degree n over K; we can for example take {x 1,...,x n} as a transcendence base.
(Quotient ring notation always uses a fraction slash " / ".) Quotient rings are distinct from the so-called "quotient field", or field of fractions, of an integral domain as well as from the more general "rings of quotients" obtained by localization.