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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_sp1.3.15.html" title="computing with radicals">Singular Book 1.3.15</a></div>
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<div><h1>Singular Book 1.3.15 -- computing with radicals</h1>
<div>Compute the radical of an ideal with <a href="_radical.html" title="the radical of an ideal">radical</a>.<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,z];</pre>
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<tr><td><pre>i2 : radical ideal(z^4+2*z^2+1)
2
o2 = ideal(- z - 1)
o2 : Ideal of R</pre>
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A somewhat more complicated example:<table class="examples"><tr><td><pre>i3 : I = ideal"xyz,x2,y4+y5"
2 5 4
o3 = ideal (x*y*z, x , y + y )
o3 : Ideal of R</pre>
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<tr><td><pre>i4 : radical I
2
o4 = ideal (-x, - y - y, x*y)
o4 : Ideal of R</pre>
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The index of nilpotency. We compute the minimal integer <i>k</i> such that <i>(y<sup>2</sup>+y)<sup>k</sup> ∈I</i>.<table class="examples"><tr><td><pre>i5 : k = 0;</pre>
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<tr><td><pre>i6 : while (y^2+y)^k % I != 0 do k = k+1;</pre>
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<tr><td><pre>i7 : k
o7 = 4</pre>
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The index of nilpotency is 4.</div>
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