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<head><title>factor(RingElement) -- factor a ring element</title>
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<div><h1>factor(RingElement) -- factor a ring element</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>factor x</tt></div>
</dd></dl>
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<li><span>Function: <a href="_factor.html" title="factor a ring element or a ZZ-module">factor</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>x</tt>, <span>a <a href="___Ring__Element.html">ring element</a></span>, or <span>a <a href="___Q__Q.html">rational number</a></span> or <span>an <a href="___Z__Z.html">integer</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Product.html">product expression</a></span>, the factorization of <tt>x</tt></span></li>
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<div class="single"><h2>Description</h2>
<div><p>The result is a <a href="___Product.html" title="the class of all product expressions">Product</a> each of whose factors is a <a href="___Power.html" title="the class of all power expressions">Power</a> whose base is one of the factors found and whose exponent is an integer.</p>
<table class="examples"><tr><td><pre>i1 : factor 124744878111332355674003415153753485211381849014286981744945
7
o1 = 3*5*53*2819 10861*10212222054939737109085868749
o1 : Expression of class Product</pre>
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<tr><td><pre>i2 : y = (2^15-4)/(2^15-5)
32764
o2 = -----
32763
o2 : QQ</pre>
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<tr><td><pre>i3 : x = factor y
2
2 8191
o3 = --------
3*67*163
o3 : Expression of class Divide</pre>
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<tr><td><pre>i4 : value x
32764
o4 = -----
32763
o4 : QQ</pre>
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<p>We may <a href="_peek.html" title="examine contents of an object">peek</a> inside <tt>x</tt> to a high depth to see its true structure as <a href="___Expression.html" title="the class of all expressions">Expression</a>.</p>
<table class="examples"><tr><td><pre>i5 : peek'(100,x)
o5 = Divide{Product{Power{2, 2}, Power{8191, 1}}, Product{Power{3, 1},
------------------------------------------------------------------------
Power{67, 1}, Power{163, 1}}}</pre>
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<p>For integers, factorization is done by <a href="_pari.html" title="">pari</a>, and the factors <i>x</i> are actually just pseudoprimes, as described in the documentation of <a href="_is__Pseudoprime_lp__Z__Z_rp.html" title="whether an integer is a pseudoprime">isPseudoprime</a>.</p>
<p>For multivariate polynomials the factorization is done with code of Michael Messollen (see <a href="___Singular-__Libfac.html" title="">Singular-Libfac</a>). For univariate polynomials the factorization is in turn done with code of Gert-Martin Greuel and Ruediger Stobbe (see <a href="___Singular-__Factory.html" title="">Singular-Factory</a>).</p>
<table class="examples"><tr><td><pre>i6 : R = ZZ/101[u]
o6 = R
o6 : PolynomialRing</pre>
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<tr><td><pre>i7 : factor (u^3-1)
2
o7 = (u - 1)(u + u + 1)
o7 : Expression of class Product</pre>
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The constant term is provided as the last factor, if it's not equal to 1.<table class="examples"><tr><td><pre>i8 : F = frac(ZZ/101[t])
o8 = F
o8 : FractionField</pre>
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<tr><td><pre>i9 : factor ((t^3-1)/(t^3+1))
2
(t - 1)(t + t + 1)
o9 = -------------------
2
(t + 1)(t - t + 1)
o9 : Expression of class Divide</pre>
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