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fraction fields

The fraction field of a ring (which must be an integral domain) is obtained with the function frac.
i1 : frac ZZ

o1 = QQ

o1 : Ring
i2 : R = ZZ/101[x,y]/(x^3 + 1 + y^3)

o2 = R

o2 : QuotientRing
i3 : frac R

o3 = frac R

o3 : FractionField
After defining a ring such as R, fractions in its fraction field can be obtained by writing them explicitly.
i4 : x

o4 = x

o4 : R
i5 : 1/x

     1
o5 = -
     x

o5 : frac R
i6 : x/1

o6 = x

o6 : R
Alternatively, after applying the function use, or assigning the fraction ring to a global variable, the symbols you used become associated with the corresponding elements of the fraction field.
i7 : use frac R

o7 = frac R

o7 : FractionField
i8 : x

o8 = x

o8 : frac R
Fractions are reduced to the extent possible. This is done by computing the syzygies between the numerator and denominator, and picking one of low degree.
i9 : f = (x-y)/(x^6-y^6)

           -1
o9 = -------------
      2          2
     x  + x*y + y

o9 : frac R
i10 : (x^3 - y^3) * f

o10 = - x + y

o10 : frac R
The parts of a fraction may be extracted.
i11 : numerator f

o11 = -1

o11 : R
i12 : denominator f

       2          2
o12 = x  + x*y + y

o12 : R
Alternatively, the functions lift and liftable can be used.
i13 : liftable(1/f,R)

o13 = true
i14 : liftable(f,R)

o14 = false
i15 : lift(1/f,R)

         2          2
o15 = - x  - x*y - y

o15 : R
Note that computations, such as Gröbner bases, over fraction fields can be quite slow.

See also