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<head><title>toField(Ring) -- declare that a ring is a field</title>
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<div><h1>toField(Ring) -- declare that a ring is a field</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>L = toField R</tt></div>
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<li><span>Function: <a href="_to__Field_lp__Ring_rp.html" title="declare that a ring is a field">toField</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>R</tt>, <span>a <a href="___Ring.html">ring</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>L</tt>, A new ring, isomorphic to <tt>R</tt>, declared to be a field. Polynomial rings over it will support Gröbner basis operations.</span></li>
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<div class="single"><h2>Description</h2>
<div><table class="examples"><tr><td><pre>i1 : A = QQ[i]/(i^2+1);</pre>
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<tr><td><pre>i2 : L = toField A
o2 = L
o2 : PolynomialRing</pre>
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<tr><td><pre>i3 : B = L[x,y,z]
o3 = B
o3 : PolynomialRing</pre>
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<tr><td><pre>i4 : I = ideal(i*x^2-y-i, i*y^2-z-i)
2 2
o4 = ideal (i*x - y - i, i*y - z - i)
o4 : Ideal of B</pre>
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<tr><td><pre>i5 : gens gb I
o5 = | y2+iz-1 x2+iy-1 |
1 2
o5 : Matrix B <--- B</pre>
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<p>If the engine eventually discovers that some nonzero element of <tt>L</tt> is not a unit, an error will be signalled. The user may then use <a href="_get__Non__Unit.html" title="retrieve a previously discovered non-unit">getNonUnit</a> to obtain a non-invertible element of <tt>L</tt>. If a ring probably is a field, it can be used as a field until a contradiction is found, and this may be a good way of discovering whether a ring is a field.</p>
<table class="examples"><tr><td><pre>i6 : A = ZZ[a]/(a^2+3);</pre>
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<tr><td><pre>i7 : L = toField A
o7 = L
o7 : PolynomialRing</pre>
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<tr><td><pre>i8 : L[x,y,z]
o8 = L[x, y, z]
o8 : PolynomialRing</pre>
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<tr><td><pre>i9 : try gb ideal (a*x^2-y^2-z^2, y^3, z^3) else getNonUnit L
o9 = a
o9 : L</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_get__Non__Unit.html" title="retrieve a previously discovered non-unit">getNonUnit</a> -- retrieve a previously discovered non-unit</span></li>
<li><span><a href="_try.html" title="catch an error">try</a> -- catch an error</span></li>
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