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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_basic_spcommutative_spalgebra.html" title="">basic commutative algebra</a> > <a href="___M2__Singular__Book.html" title="Macaulay2 examples for the Singular book">M2SingularBook</a> > <a href="___Singular_sp__Book_sp1.1.8.html" title="computation in fields">Singular Book 1.1.8</a></div>
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<div><h1>Singular Book 1.1.8 -- computation in fields</h1>
<div><h2>Computation over ZZ and QQ</h2>
In Macaulay2, Integers are arbitary precision. The ring of integers is denoted ZZ.<table class="examples"><tr><td><pre>i1 : 123456789^5
o1 = 28679718602997181072337614380936720482949</pre>
</td></tr>
<tr><td><pre>i2 : matrix{{123456789^5}}
o2 = | 28679718602997181072337614380936720482949 |
1 1
o2 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i3 : gcd(3782621293644611237896400,85946734897630958700)
o3 = 100</pre>
</td></tr>
</table>
The ring of rational numbers is denoted by QQ.<table class="examples"><tr><td><pre>i4 : n = 12345/6789
4115
o4 = ----
2263
o4 : QQ</pre>
</td></tr>
<tr><td><pre>i5 : n^5
1179910858126071875
o5 = -------------------
59350279669807543
o5 : QQ</pre>
</td></tr>
<tr><td><pre>i6 : toString(n^5)
o6 = 1179910858126071875/59350279669807543</pre>
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</table>
<h2>Computation in finite fields</h2>
<table class="examples"><tr><td><pre>i7 : A = ZZ/32003;</pre>
</td></tr>
</table>
In order to do arithmetic in this ring, you must construct elements of this ring. <tt>n_A</tt> gives the image of the integer n in A.<table class="examples"><tr><td><pre>i8 : 123456789 * 1_A
o8 = -10785
o8 : A</pre>
</td></tr>
<tr><td><pre>i9 : (123456789_A)^5
o9 = 8705
o9 : A</pre>
</td></tr>
</table>
<table class="examples"><tr><td><pre>i10 : A2 = GF(8,Variable=>a)
o10 = A2
o10 : GaloisField</pre>
</td></tr>
<tr><td><pre>i11 : ambient A2
ZZ
--[a]
2
o11 = ----------
3
a + a + 1
o11 : QuotientRing</pre>
</td></tr>
<tr><td><pre>i12 : a^3+a+1
o12 = 0
o12 : A2</pre>
</td></tr>
</table>
<table class="examples"><tr><td><pre>i13 : A3 = ZZ/2[a]/(a^20+a^3+1);</pre>
</td></tr>
<tr><td><pre>i14 : n = a+a^2
2
o14 = a + a
o14 : A3</pre>
</td></tr>
<tr><td><pre>i15 : n^5
10 9 6 5
o15 = a + a + a + a
o15 : A3</pre>
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</table>
<h2>Computing with real and complex numbers</h2>
<table class="examples"><tr><td><pre>i16 : n = 123456789.0
o16 = 123456789
o16 : RR (of precision 53)</pre>
</td></tr>
<tr><td><pre>i17 : n = n * 1_RR
o17 = 123456789
o17 : RR (of precision 53)</pre>
</td></tr>
<tr><td><pre>i18 : n^5
o18 = 2.86797186029972e40
o18 : RR (of precision 53)</pre>
</td></tr>
</table>
<h2>Computing with parameters</h2>
<table class="examples"><tr><td><pre>i19 : R3 = frac(ZZ[a,b,c])
o19 = R3
o19 : FractionField</pre>
</td></tr>
<tr><td><pre>i20 : n = 12345*a + 12345/(78*b*c)
320970a*b*c + 4115
o20 = ------------------
26b*c
o20 : R3</pre>
</td></tr>
<tr><td><pre>i21 : n^2
2 2 2
103021740900a b c + 2641583100a*b*c + 16933225
o21 = -----------------------------------------------
2 2
676b c
o21 : R3</pre>
</td></tr>
<tr><td><pre>i22 : n/(9*c)
320970a*b*c + 4115
o22 = ------------------
2
234b*c
o22 : R3</pre>
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