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<div><h1>Singular Book 1.8.9 -- radical membership</h1>
<div>Recall that an element <i>f</i> is in an ideal <i>I</i> if <i>1 ∈(I, tf-1) ⊂R[t]</i>.<table class="examples"><tr><td><pre>i1 : A = QQ[x,y,z];</pre>
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<tr><td><pre>i2 : I = ideal"x5,xy3,y7,z3+xyz";
o2 : Ideal of A</pre>
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<tr><td><pre>i3 : f = x+y+z;</pre>
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<table class="examples"><tr><td><pre>i4 : B = A[t];</pre>
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<tr><td><pre>i5 : J = substitute(I,B) + ideal(f*t-1)
5 3 7 3
o5 = ideal (x , x*y , y , x*y*z + z , (x + y + z)t - 1)
o5 : Ideal of B</pre>
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<tr><td><pre>i6 : 1 % J
o6 = 0
o6 : B</pre>
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The polynomial f is in the radical. Let's compute the radical to make sure.<table class="examples"><tr><td><pre>i7 : radical I
o7 = ideal (z, y, x)
o7 : Ideal of A</pre>
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