/usr/share/gap/doc/ref/chap68.html is in gap-doc 4r6p5-3.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 | <?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (ref) - Chapter 68: p-adic Numbers (preliminary)</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap68" onload="jscontent()">
<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chap10.html">10</a> <a href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chap14.html">14</a> <a href="chap15.html">15</a> <a href="chap16.html">16</a> <a href="chap17.html">17</a> <a href="chap18.html">18</a> <a href="chap19.html">19</a> <a href="chap20.html">20</a> <a href="chap21.html">21</a> <a href="chap22.html">22</a> <a href="chap23.html">23</a> <a href="chap24.html">24</a> <a href="chap25.html">25</a> <a href="chap26.html">26</a> <a href="chap27.html">27</a> <a href="chap28.html">28</a> <a href="chap29.html">29</a> <a href="chap30.html">30</a> <a href="chap31.html">31</a> <a href="chap32.html">32</a> <a href="chap33.html">33</a> <a href="chap34.html">34</a> <a href="chap35.html">35</a> <a href="chap36.html">36</a> <a href="chap37.html">37</a> <a href="chap38.html">38</a> <a href="chap39.html">39</a> <a href="chap40.html">40</a> <a href="chap41.html">41</a> <a href="chap42.html">42</a> <a href="chap43.html">43</a> <a href="chap44.html">44</a> <a href="chap45.html">45</a> <a href="chap46.html">46</a> <a href="chap47.html">47</a> <a href="chap48.html">48</a> <a href="chap49.html">49</a> <a href="chap50.html">50</a> <a href="chap51.html">51</a> <a href="chap52.html">52</a> <a href="chap53.html">53</a> <a href="chap54.html">54</a> <a href="chap55.html">55</a> <a href="chap56.html">56</a> <a href="chap57.html">57</a> <a href="chap58.html">58</a> <a href="chap59.html">59</a> <a href="chap60.html">60</a> <a href="chap61.html">61</a> <a href="chap62.html">62</a> <a href="chap63.html">63</a> <a href="chap64.html">64</a> <a href="chap65.html">65</a> <a href="chap66.html">66</a> <a href="chap67.html">67</a> <a href="chap68.html">68</a> <a href="chap69.html">69</a> <a href="chap70.html">70</a> <a href="chap71.html">71</a> <a href="chap72.html">72</a> <a href="chap73.html">73</a> <a href="chap74.html">74</a> <a href="chap75.html">75</a> <a href="chap76.html">76</a> <a href="chap77.html">77</a> <a href="chap78.html">78</a> <a href="chap79.html">79</a> <a href="chap80.html">80</a> <a href="chap81.html">81</a> <a href="chap82.html">82</a> <a href="chap83.html">83</a> <a href="chap84.html">84</a> <a href="chap85.html">85</a> <a href="chap86.html">86</a> <a href="chap87.html">87</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>
<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap67.html">[Previous Chapter]</a> <a href="chap69.html">[Next Chapter]</a> </div>
<p id="mathjaxlink" class="pcenter"><a href="chap68_mj.html">[MathJax on]</a></p>
<p><a id="X7C6B3CBB873253E3" name="X7C6B3CBB873253E3"></a></p>
<div class="ChapSects"><a href="chap68.html#X7C6B3CBB873253E3">68 <span class="Heading">p-adic Numbers (preliminary)</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap68.html#X7F81667C81655050">68.1 <span class="Heading">Pure p-adic Numbers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap68.html#X82D1AD1D872B480D">68.1-1 PurePadicNumberFamily</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap68.html#X84A79ED87B47CC07">68.1-2 PadicNumber</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap68.html#X80D67BB67A509A56">68.1-3 Valuation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap68.html#X79059A9E876C8198">68.1-4 ShiftedPadicNumber</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap68.html#X7AD7FA3786AF9F0E">68.1-5 IsPurePadicNumber</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap68.html#X83B2BA4586ECAA5C">68.1-6 IsPurePadicNumberFamily</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap68.html#X83EEF8197D212075">68.2 <span class="Heading">Extensions of the p-adic Numbers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap68.html#X83EE630D7885DB3D">68.2-1 PadicExtensionNumberFamily</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap68.html#X7C6F2F018084AFC4">68.2-2 PadicNumber</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap68.html#X7923FC147BDCC810">68.2-3 IsPadicExtensionNumber</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap68.html#X868807D487DAF713">68.2-4 IsPadicExtensionNumberFamily</a></span>
</div></div>
</div>
<h3>68 <span class="Heading">p-adic Numbers (preliminary)</span></h3>
<p>In this chapter <span class="SimpleMath">p</span> is always a (fixed) prime integer.</p>
<p>The <span class="SimpleMath">p</span>-adic numbers <span class="SimpleMath">Q_p</span> are the completion of the rational numbers with respect to the valuation <span class="SimpleMath">ν_p( p^v ⋅ a / b) = v</span> if <span class="SimpleMath">p</span> divides neither <span class="SimpleMath">a</span> nor <span class="SimpleMath">b</span>. They form a field of characteristic 0 which nevertheless shows some behaviour of the finite field with <span class="SimpleMath">p</span> elements.</p>
<p>A <span class="SimpleMath">p</span>-adic numbers can be represented by a "<span class="SimpleMath">p</span>-adic expansion" which is similar to the decimal expansion used for the reals (but written from left to right). So for example if <span class="SimpleMath">p = 2</span>, the numbers <span class="SimpleMath">1</span>, <span class="SimpleMath">2</span>, <span class="SimpleMath">3</span>, <span class="SimpleMath">4</span>, <span class="SimpleMath">1/2</span>, and <span class="SimpleMath">4/5</span> are represented as <span class="SimpleMath">1(2)</span>, <span class="SimpleMath">0.1(2)</span>, <span class="SimpleMath">1.1(2)</span>, <span class="SimpleMath">0.01(2)</span>, <span class="SimpleMath">10(2)</span>, and the infinite periodic expansion <span class="SimpleMath">0.010110011001100...(2)</span>. <span class="SimpleMath">p</span>-adic numbers can be approximated by ignoring higher powers of <span class="SimpleMath">p</span>, so for example with only 2 digits accuracy <span class="SimpleMath">4/5</span> would be approximated as <span class="SimpleMath">0.01(2)</span>. This is different from the decimal approximation of real numbers in that <span class="SimpleMath">p</span>-adic approximation is a ring homomorphism on the subrings of <span class="SimpleMath">p</span>-adic numbers whose valuation is bounded from below so that rounding errors do not increase with repeated calculations.</p>
<p>In <strong class="pkg">GAP</strong>, <span class="SimpleMath">p</span>-adic numbers are always represented by such approximations. A family of approximated <span class="SimpleMath">p</span>-adic numbers consists of <span class="SimpleMath">p</span>-adic numbers with a fixed prime <span class="SimpleMath">p</span> and a certain precision, and arithmetic with these numbers is done with this precision.</p>
<p><a id="X7F81667C81655050" name="X7F81667C81655050"></a></p>
<h4>68.1 <span class="Heading">Pure p-adic Numbers</span></h4>
<p>Pure <span class="SimpleMath">p</span>-adic numbers are the <span class="SimpleMath">p</span>-adic numbers described so far.</p>
<p><a id="X82D1AD1D872B480D" name="X82D1AD1D872B480D"></a></p>
<h5>68.1-1 PurePadicNumberFamily</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PurePadicNumberFamily</code>( <var class="Arg">p</var>, <var class="Arg">precision</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the family of pure <span class="SimpleMath">p</span>-adic numbers over the prime <var class="Arg">p</var> with <var class="Arg">precision</var> "digits". That is to say, the approximate value will differ from the correct value by a multiple of <span class="SimpleMath">p^digits</span>.</p>
<p><a id="X84A79ED87B47CC07" name="X84A79ED87B47CC07"></a></p>
<h5>68.1-2 PadicNumber</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PadicNumber</code>( <var class="Arg">fam</var>, <var class="Arg">rat</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the element of the <span class="SimpleMath">p</span>-adic number family <var class="Arg">fam</var> that approximates the rational number <var class="Arg">rat</var>.</p>
<p><span class="SimpleMath">p</span>-adic numbers allow the usual operations for fields.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">fam:=PurePadicNumberFamily(2,20);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=PadicNumber(fam,4/5);</span>
0.010110011001100110011(2)
<span class="GAPprompt">gap></span> <span class="GAPinput">fam:=PurePadicNumberFamily(2,3);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=PadicNumber(fam,4/5);</span>
0.0101(2)
<span class="GAPprompt">gap></span> <span class="GAPinput">3*a;</span>
0.0111(2)
<span class="GAPprompt">gap></span> <span class="GAPinput">a/2;</span>
0.101(2)
<span class="GAPprompt">gap></span> <span class="GAPinput">a*10;</span>
0.001(2)
</pre></div>
<p>See <code class="func">PadicNumber</code> (<a href="chap68.html#X7C6F2F018084AFC4"><span class="RefLink">68.2-2</span></a>) for other methods for <code class="func">PadicNumber</code>.</p>
<p><a id="X80D67BB67A509A56" name="X80D67BB67A509A56"></a></p>
<h5>68.1-3 Valuation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Valuation</code>( <var class="Arg">obj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The valuation is the <span class="SimpleMath">p</span>-part of the <span class="SimpleMath">p</span>-adic number.</p>
<p><a id="X79059A9E876C8198" name="X79059A9E876C8198"></a></p>
<h5>68.1-4 ShiftedPadicNumber</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ShiftedPadicNumber</code>( <var class="Arg">padic</var>, <var class="Arg">int</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">ShiftedPadicNumber</code> takes a <span class="SimpleMath">p</span>-adic number <var class="Arg">padic</var> and an integer <var class="Arg">shift</var> and returns the <span class="SimpleMath">p</span>-adic number <span class="SimpleMath">c</span>, that is <var class="Arg">padic</var> <code class="code">*</code> <span class="SimpleMath">p</span><code class="code">^</code><var class="Arg">shift</var>.</p>
<p><a id="X7AD7FA3786AF9F0E" name="X7AD7FA3786AF9F0E"></a></p>
<h5>68.1-5 IsPurePadicNumber</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPurePadicNumber</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>The category of pure <span class="SimpleMath">p</span>-adic numbers.</p>
<p><a id="X83B2BA4586ECAA5C" name="X83B2BA4586ECAA5C"></a></p>
<h5>68.1-6 IsPurePadicNumberFamily</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPurePadicNumberFamily</code>( <var class="Arg">fam</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>The family of pure <span class="SimpleMath">p</span>-adic numbers.</p>
<p><a id="X83EEF8197D212075" name="X83EEF8197D212075"></a></p>
<h4>68.2 <span class="Heading">Extensions of the p-adic Numbers</span></h4>
<p>The usual Kronecker construction with an irreducible polynomial can be used to construct extensions of the <span class="SimpleMath">p</span>-adic numbers. Let <span class="SimpleMath">L</span> be such an extension. Then there is a subfield <span class="SimpleMath">K < L</span> such that <span class="SimpleMath">K</span> is an unramified extension of the <span class="SimpleMath">p</span>-adic numbers and <span class="SimpleMath">L/K</span> is purely ramified.</p>
<p>(For an explanation of "ramification" see for example <a href="chapBib.html#biBneukirch">[Neu92, Section II.7]</a>, or another book on algebraic number theory. Essentially, an extension <span class="SimpleMath">L</span> of the <span class="SimpleMath">p</span>-adic numbers generated by a rational polynomial <span class="SimpleMath">f</span> is unramified if <span class="SimpleMath">f</span> remains squarefree modulo <span class="SimpleMath">p</span> and is completely ramified if modulo <span class="SimpleMath">p</span> the polynomial <span class="SimpleMath">f</span> is a power of a linear factor while remaining irreducible over the <span class="SimpleMath">p</span>-adic numbers.)</p>
<p>The representation of extensions of <span class="SimpleMath">p</span>-adic numbers in <strong class="pkg">GAP</strong> uses the subfield <span class="SimpleMath">K</span>.</p>
<p><a id="X83EE630D7885DB3D" name="X83EE630D7885DB3D"></a></p>
<h5>68.2-1 PadicExtensionNumberFamily</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PadicExtensionNumberFamily</code>( <var class="Arg">p</var>, <var class="Arg">precision</var>, <var class="Arg">unram</var>, <var class="Arg">ram</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>An extended <span class="SimpleMath">p</span>-adic field <span class="SimpleMath">L</span> is given by two polynomials <span class="SimpleMath">h</span> and <span class="SimpleMath">g</span> with coefficient lists <var class="Arg">unram</var> (for the unramified part) and <var class="Arg">ram</var> (for the ramified part). Then <span class="SimpleMath">L</span> is isomorphic to <span class="SimpleMath">Q_p[x,y]/(h(x),g(y))</span>.</p>
<p>This function takes the prime number <var class="Arg">p</var> and the two coefficient lists <var class="Arg">unram</var> and <var class="Arg">ram</var> for the two polynomials. The polynomial given by the coefficients in <var class="Arg">unram</var> must be a cyclotomic polynomial and the polynomial given by <var class="Arg">ram</var> must be either an Eisenstein polynomial or <span class="SimpleMath">1+x</span>. <em>This is not checked by <strong class="pkg">GAP</strong>.</em></p>
<p>Every number in <span class="SimpleMath">L</span> is represented as a coefficient list w. r. t. the basis <span class="SimpleMath">{ 1, x, x^2, ..., y, xy, x^2 y, ... }</span> of <span class="SimpleMath">L</span>. The integer <var class="Arg">precision</var> is the number of "digits" that all the coefficients have.</p>
<p><em>A general comment:</em></p>
<p>The polynomials with which <code class="func">PadicExtensionNumberFamily</code> is called define an extension of <span class="SimpleMath">Q_p</span>. It must be ensured that both polynomials are really irreducible over <span class="SimpleMath">Q_p</span>! For example <span class="SimpleMath">x^2+x+1</span> is <em>not</em> irreducible over <span class="SimpleMath">Q_p</span>. Therefore the "extension" <code class="code">PadicExtensionNumberFamily(3, 4, [1,1,1], [1,1])</code> contains non-invertible "pseudo-p-adic numbers". Conversely, if an "extension" contains noninvertible elements then one of the defining polynomials was not irreducible.</p>
<p><a id="X7C6F2F018084AFC4" name="X7C6F2F018084AFC4"></a></p>
<h5>68.2-2 PadicNumber</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PadicNumber</code>( <var class="Arg">fam</var>, <var class="Arg">rat</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PadicNumber</code>( <var class="Arg">purefam</var>, <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PadicNumber</code>( <var class="Arg">extfam</var>, <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>(see also <code class="func">PadicNumber</code> (<a href="chap68.html#X84A79ED87B47CC07"><span class="RefLink">68.1-2</span></a>)).</p>
<p><code class="func">PadicNumber</code> creates a <span class="SimpleMath">p</span>-adic number in the <span class="SimpleMath">p</span>-adic numbers family <var class="Arg">fam</var>. The first form returns the <span class="SimpleMath">p</span>-adic number corresponding to the rational <var class="Arg">rat</var>.</p>
<p>The second form takes a pure <span class="SimpleMath">p</span>-adic numbers family <var class="Arg">purefam</var> and a list <var class="Arg">list</var> of length two, and returns the number <span class="SimpleMath">p</span><code class="code">^</code><var class="Arg">list</var><code class="code">[1] * </code><var class="Arg">list</var><code class="code">[2]</code>. It must be guaranteed that no entry of <var class="Arg">list</var><code class="code">[2]</code> is divisible by the prime <span class="SimpleMath">p</span>. (Otherwise precision will get lost.)</p>
<p>The third form creates a number in the family <var class="Arg">extfam</var> of a <span class="SimpleMath">p</span>-adic extension. The second argument must be a list <var class="Arg">list</var> of length two such that <var class="Arg">list</var><code class="code">[2]</code> is the list of coefficients w.r.t. the basis <span class="SimpleMath">{ 1, ..., x^{f-1} ⋅ y^{e-1} }</span> of the extended <span class="SimpleMath">p</span>-adic field and <var class="Arg">list</var><code class="code">[1]</code> is a common <span class="SimpleMath">p</span>-part of all these coefficients.</p>
<p><span class="SimpleMath">p</span>-adic numbers admit the usual field operations.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">efam:=PadicExtensionNumberFamily(3, 5, [1,1,1], [1,1]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PadicNumber(efam,7/9);</span>
padic(120(3),0(3))
</pre></div>
<p><em>A word of warning:</em></p>
<p>Depending on the actual representation of quotients, precision may seem to "vanish". For example in <code class="code">PadicExtensionNumberFamily(3, 5, [1,1,1], [1,1])</code> the number <code class="code">(1.2000, 0.1210)(3)</code> can be represented as <code class="code">[ 0, [ 1.2000, 0.1210 ] ]</code> or as <code class="code">[ -1, [ 12.000, 1.2100 ] ]</code> (here the coefficients have to be multiplied by <span class="SimpleMath">p^{-1}</span>).</p>
<p>So there may be a number <code class="code">(1.2, 2.2)(3)</code> which seems to have only two digits of precision instead of the declared 5. But internally the number is stored as <code class="code">[ -3, [ 0.0012, 0.0022 ] ]</code> and so has in fact maximum precision.</p>
<p><a id="X7923FC147BDCC810" name="X7923FC147BDCC810"></a></p>
<h5>68.2-3 IsPadicExtensionNumber</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPadicExtensionNumber</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>The category of elements of the extended <span class="SimpleMath">p</span>-adic field.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput"> efam:=PadicExtensionNumberFamily(3, 5, [1,1,1], [1,1]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPadicExtensionNumber(PadicNumber(efam,7/9));</span>
true
</pre></div>
<p><a id="X868807D487DAF713" name="X868807D487DAF713"></a></p>
<h5>68.2-4 IsPadicExtensionNumberFamily</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPadicExtensionNumberFamily</code>( <var class="Arg">fam</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Family of elements of the extended <span class="SimpleMath">p</span>-adic field.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">efam:=PadicExtensionNumberFamily(3, 5, [1,1,1], [1,1]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPadicExtensionNumberFamily(efam);</span>
true
</pre></div>
<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap67.html">[Previous Chapter]</a> <a href="chap69.html">[Next Chapter]</a> </div>
<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chap10.html">10</a> <a href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chap14.html">14</a> <a href="chap15.html">15</a> <a href="chap16.html">16</a> <a href="chap17.html">17</a> <a href="chap18.html">18</a> <a href="chap19.html">19</a> <a href="chap20.html">20</a> <a href="chap21.html">21</a> <a href="chap22.html">22</a> <a href="chap23.html">23</a> <a href="chap24.html">24</a> <a href="chap25.html">25</a> <a href="chap26.html">26</a> <a href="chap27.html">27</a> <a href="chap28.html">28</a> <a href="chap29.html">29</a> <a href="chap30.html">30</a> <a href="chap31.html">31</a> <a href="chap32.html">32</a> <a href="chap33.html">33</a> <a href="chap34.html">34</a> <a href="chap35.html">35</a> <a href="chap36.html">36</a> <a href="chap37.html">37</a> <a href="chap38.html">38</a> <a href="chap39.html">39</a> <a href="chap40.html">40</a> <a href="chap41.html">41</a> <a href="chap42.html">42</a> <a href="chap43.html">43</a> <a href="chap44.html">44</a> <a href="chap45.html">45</a> <a href="chap46.html">46</a> <a href="chap47.html">47</a> <a href="chap48.html">48</a> <a href="chap49.html">49</a> <a href="chap50.html">50</a> <a href="chap51.html">51</a> <a href="chap52.html">52</a> <a href="chap53.html">53</a> <a href="chap54.html">54</a> <a href="chap55.html">55</a> <a href="chap56.html">56</a> <a href="chap57.html">57</a> <a href="chap58.html">58</a> <a href="chap59.html">59</a> <a href="chap60.html">60</a> <a href="chap61.html">61</a> <a href="chap62.html">62</a> <a href="chap63.html">63</a> <a href="chap64.html">64</a> <a href="chap65.html">65</a> <a href="chap66.html">66</a> <a href="chap67.html">67</a> <a href="chap68.html">68</a> <a href="chap69.html">69</a> <a href="chap70.html">70</a> <a href="chap71.html">71</a> <a href="chap72.html">72</a> <a href="chap73.html">73</a> <a href="chap74.html">74</a> <a href="chap75.html">75</a> <a href="chap76.html">76</a> <a href="chap77.html">77</a> <a href="chap78.html">78</a> <a href="chap79.html">79</a> <a href="chap80.html">80</a> <a href="chap81.html">81</a> <a href="chap82.html">82</a> <a href="chap83.html">83</a> <a href="chap84.html">84</a> <a href="chap85.html">85</a> <a href="chap86.html">86</a> <a href="chap87.html">87</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>
<hr />
<p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>
|