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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_rings.html" title="">rings</a> > <a href="_fraction_spfields.html" title="">fraction fields</a></div>
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<div><h1>fraction fields</h1>
<div>The fraction field of a ring (which must be an integral domain) is obtained with the function <a href="_frac.html" title="construct a fraction field">frac</a>.<table class="examples"><tr><td><pre>i1 : frac ZZ
o1 = QQ
o1 : Ring</pre>
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<tr><td><pre>i2 : R = ZZ/101[x,y]/(x^3 + 1 + y^3)
o2 = R
o2 : QuotientRing</pre>
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<tr><td><pre>i3 : frac R
o3 = frac(R)
o3 : FractionField</pre>
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After defining a ring such as <tt>R</tt>, fractions in its fraction field can be obtained by writing them explicitly.<table class="examples"><tr><td><pre>i4 : x
o4 = x
o4 : R</pre>
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<tr><td><pre>i5 : 1/x
1
o5 = -
x
o5 : frac(R)</pre>
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<tr><td><pre>i6 : x/1
o6 = x
o6 : R</pre>
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</table>
Alternatively, after applying the function <a href="_use.html" title="install or activate object">use</a>, or assigning the fraction ring to a global variable, the symbols you used become associated with the corresponding elements of the fraction field.<table class="examples"><tr><td><pre>i7 : use frac R
o7 = frac(R)
o7 : FractionField</pre>
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<tr><td><pre>i8 : x
o8 = x
o8 : frac(R)</pre>
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</table>
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.<table class="examples"><tr><td><pre>i9 : f = (x-y)/(x^6-y^6)
-1
o9 = -------------
2 2
x + x*y + y
o9 : frac(R)</pre>
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<tr><td><pre>i10 : (x^3 - y^3) * f
o10 = - x + y
o10 : frac(R)</pre>
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The parts of a fraction may be extracted.<table class="examples"><tr><td><pre>i11 : numerator f
o11 = -1
o11 : R</pre>
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<tr><td><pre>i12 : denominator f
2 2
o12 = x + x*y + y
o12 : R</pre>
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Alternatively, the functions <a href="_lift.html" title="lift to another ring">lift</a> and <a href="_liftable.html" title="whether lifting to another ring is possible">liftable</a> can be used.<table class="examples"><tr><td><pre>i13 : liftable(1/f,R)
o13 = true</pre>
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<tr><td><pre>i14 : liftable(f,R)
o14 = false</pre>
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<tr><td><pre>i15 : lift(1/f,R)
2 2
o15 = - x - x*y - y
o15 : R</pre>
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Note that computations, such as Gröbner bases, over fraction fields can be quite slow.</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_frac.html" title="construct a fraction field">frac</a> -- construct a fraction field</span></li>
<li><span><a href="_numerator.html" title="numerator of a fraction">numerator</a> -- numerator of a fraction</span></li>
<li><span><a href="_denominator.html" title="denominator of a fraction">denominator</a> -- denominator of a fraction</span></li>
<li><span><a href="_liftable.html" title="whether lifting to another ring is possible">liftable</a> -- whether lifting to another ring is possible</span></li>
<li><span><a href="_lift.html" title="lift to another ring">lift</a> -- lift to another ring</span></li>
<li><span><a href="_kernel_lp__Ring__Map_rp.html" title="kernel of a ringmap">kernel(RingMap)</a> -- kernel of a ringmap</span></li>
</ul>
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