<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- -->
<!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html -->
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" >
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head><title>GF -- make a finite field</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_global.html">next</a> | <a href="_get__W__W__W.html">previous</a> | <a href="_global.html">forward</a> | <a href="_get__W__W__W.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div>
</td>
</tr>
</table>
<hr/>
<div><h1>GF -- make a finite field</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="___G__F.html">PrimitiveElement => ...</a>, -- make a finite field</span></li>
<li><span><a href="___G__F.html">SizeLimit => ...</a>, -- make a finite field</span></li>
<li><span><a href="___G__F.html">Variable => ...</a>, -- make a finite field</span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div><div><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>GF(p,n)</tt><br/><tt>GF(q)</tt></div>
</dd></dl>
</div>
</li>
<li>Inputs:<ul><li><span><tt>p</tt>, a prime number</span></li>
<li><span><tt>n</tt></span></li>
<li><span><tt>Variable => </tt><span><span>a <a href="___Symbol.html">symbol</a></span>, <span>default value null</span>, the name to use for the generator of the field</span></span></li>
<li><span><tt>SizeLimit => </tt><span><span>an <a href="___Z__Z.html">integer</a></span>, <span>default value 10000</span>, the limit on the size of a Galois field whose elements will be represented internally as powers of the primitive element</span></span></li>
</ul>
</li>
<li>Outputs:<ul><li><span><span>a <a href="___Galois__Field.html">Galois field</a></span>, a finite field with <tt>q = p^n</tt> elements</span></li>
</ul>
</li>
</ul>
The generator of this ring is a primitive element: it generates the multiplicative group of non-zero elements.<p>If the single argument form GF(q) is given, q should be a prime power q = p^n</p>
<table class="examples"><tr><td><pre>i1 : A = GF(3,2,Variable=>b);</pre>
</td></tr>
<tr><td><pre>i2 : ambient A
ZZ
--[b]
3
o2 = ----------
2
b - b - 1
o2 : QuotientRing</pre>
</td></tr>
<tr><td><pre>i3 : b^8
o3 = 1
o3 : A</pre>
</td></tr>
<tr><td><pre>i4 : b^4
o4 = -1
o4 : A</pre>
</td></tr>
</table>
<table class="examples"><tr><td><pre>i5 : K = GF 8
o5 = K
o5 : GaloisField</pre>
</td></tr>
<tr><td><pre>i6 : x = K_0
o6 = a
o6 : K</pre>
</td></tr>
<tr><td><pre>i7 : x^3+x
o7 = 1
o7 : K</pre>
</td></tr>
</table>
</div>
<div><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>GF R</tt></div>
</dd></dl>
</div>
</li>
<li>Inputs:<ul><li><span><tt>R</tt>, <span>a <a href="___Ring.html">ring</a></span>, A quotient of a polynomial ring over <tt>ZZ/p</tt> in one variable, modulo an irreducible polynomial</span></li>
<li><span><tt>PrimitiveElement => </tt><span><span>default value FindOne</span>, either an element of <tt>R</tt>, or the symbol <a href="___Find__One.html" title="find a primitive element">FindOne</a>. An element is primitive if it generates the multiplicative group of non-zero elements of R</span></span></li>
<li><span><tt>SizeLimit => </tt><span><span>an <a href="___Z__Z.html">integer</a></span>, <span>default value 10000</span>, the limit on the size of a Galois field whose elements will be represented internally as powers of the primitive element</span></span></li>
</ul>
</li>
<li>Outputs:<ul><li><span><span>a <a href="___Galois__Field.html">Galois field</a></span>, a finite field isomorphic to <tt>R</tt></span></li>
</ul>
</li>
</ul>
<table class="examples"><tr><td><pre>i8 : A = ZZ/5[a]/(a^3-a-2)
o8 = A
o8 : QuotientRing</pre>
</td></tr>
<tr><td><pre>i9 : B = GF A
o9 = B
o9 : GaloisField</pre>
</td></tr>
<tr><td><pre>i10 : C = ZZ/5[b]/(b^3+1+3*b^2+b)
o10 = C
o10 : QuotientRing</pre>
</td></tr>
<tr><td><pre>i11 : D = GF C
o11 = D
o11 : GaloisField</pre>
</td></tr>
<tr><td><pre>i12 : map(B,D,{a^2})
2
o12 = map(B,D,{a })
o12 : RingMap B <--- D</pre>
</td></tr>
</table>
</div>
</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_to__Field_lp__Ring_rp.html" title="declare that a ring is a field">toField</a> -- declare that a ring is a field</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>GF</tt> :</h2>
<ul><li>GF(Ring)</li>
<li>GF(ZZ)</li>
<li>GF(ZZ,ZZ)</li>
</ul>
</div>
</div>
</body>
</html>
