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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.
The function field of the n-dimensional space over a field F is F(x 1, ..., x n), i.e., the field consisting of ratios of polynomials in n indeterminates. The function field of X is the same as the one of any open dense subvariety. In other words, the function field is insensitive to replacing X by a (slightly) smaller subvariety.
To give a non-example, [4] let k be a field and = [,] [], the subalgebra generated by t 2 and t 3. Then A is not integrally closed: it has the field of fractions k ( t ) {\displaystyle k(t)} , and the monic polynomial X 2 − t 2 {\displaystyle X^{2}-t^{2}} in the variable X has root t which is in the field of fractions but not in A.
Let be an integral domain, and let = be its field of fractions.. A fractional ideal of is an -submodule of such that there exists a non-zero such that .The element can be thought of as clearing out the denominators in , hence the name fractional ideal.
In fact, the fraction fields of the rings of regular functions on any affine open set will be the same, so we define, for any U, K X (U) to be the common fraction field of any ring of regular functions on any open affine subset of X. Alternatively, one can define the function field in this case to be the local ring of the generic point.
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers ; they may be taken in any field K .
For example, if = then its function field is isomorphic to () where is an indeterminant and the field is the field of fractions of polynomials in . Then, a place v p {\displaystyle v_{p}} at a point p ∈ X {\displaystyle p\in X} measures the order of vanishing or the order of a pole of a fraction of polynomials p ( x ) / q ( x ) {\displaystyle ...
The algebraic function fields over k form a category; the morphisms from function field K to L are the ring homomorphisms f : K → L with f(a) = a for all a in k. All these morphisms are injective. If K is a function field over k of n variables, and L is a function field in m variables, and n > m, then there are no morphisms from K to L.