|
[____]
Given a ring R such that there is a greatest-common-divisor algorithm for
polynomials over R, Magma allows the construction of a rational
function field K in any number of indeterminates over R.
Such function fields are objects of type FldFunRat with elements
of type FldFunRatElt.
The elements of K are fractions whose numerators and denominators lie
in the corresponding polynomial ring over R.
As for polynomial rings, the different
univariate and multivariate cases are distinguished, since the fractions
just use the different representations given by the different cases of
polynomial rings.
A fraction f lying in a function field K is always reduced; this
means that the numerator and denominator of f are
coprime and the denominator of
f is normalized
(monic over fields and positive over Z).
Note that R itself need not be a field. Thus it is possible, for
example, to create the rational function field K = Z(t) which is
mathematically equal to Q(t) of course, but will be represented
slightly differently. A fraction in Q(t) will have a monic denominator
(and the coefficients of both the numerator and denominator may be
non-integral), while a fraction in Z(t) will have a positive denominator
(and the coefficients of both the numerator and denominator will be integral).
Thus the
fractions (3t + 2)/(4t - 2) ∈Z(t) and ((3/4)t + 1/2)/(t - 1/2)
∈Q(t) are equal and are both reduced in their respective fields.
It is generally much better to use the domain of integers instead of the
field of fractions for the coefficient ring R (so it is better to
use Z(t) instead of Q(t)) since arithmetic is much faster, but
the use of a field of fractions for the coefficient ring may be more
desirable for output purposes.
[Next][Prev] [Right] [____] [Up] [Index] [Root]
|