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<html xmlns="http://www.w3.org/1999/xhtml"<>f the first argument <var class="Arg">wfilt</var> is given, then it must be either <code class="code">IsSyllableWordsFamily</code>, <code class="code">IsLetterWordsFamily</code>, <code class="code">IsWLetterWordsFamily</code>, or <code class="code">IsBLetterWordsFamily</.  java.lang.StringIndexOutOfBoundsException: Range [313, 312) out of bounds for length 521
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<class"hlinktop<span "">oto <<chap0html>Top</>< hrefc.html></  href"chap2.html"2  a href="hap3html"3</   hrefchap4.">/><a =.html<a  < href.html>/> <chap7h"<>ahref".>/><java.lang.StringIndexOutOfBoundsException: Range [323, 322) out of bounds for length 2758

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<div class="ChapSects="ChapSects>< =hap68html> <class""p ()/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="span class="ContSS"><br /><span class="nocss">  </span><a href="chap68.html#X80D67BB67A509A56">68.1-3 Valuation</a></span>
< /java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 133
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap68.html#X79059A9E876C8198">68.1-4 ShiftedPadicNumber</a></span>
< java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 141
<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 147
</div></java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 0
<divclass"ontSect">span class="ocline">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">< />span class=nocss"&;nbsp;/>a =".#X83EE630D7885DB3D682--1 PadicExtensionNumberFamily<<a</pan>
<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>
</iv</div>
</div>

<h3>68 <span class="Heading">p-adic Numbers (preliminary)</span></h3>

<p>In this chapter <span class="SimpleMath"><div</div>

<p>The

<p>A <span class="SimpleMath">p</span>-adic numbers can be represented by a <q><span class="SimpleMathjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>In <strong class="pkg">GAP</strong>, <span class="SimpleMath">p</span>-adic numbers are always

<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.</java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<d="D1AD1D872B480D "82<>p

<<idX java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 62

<iv class==func"><table class="func"width=100"<>< class="dleft"< class=func"#; PurePadicNumberFamily</(<var =""></ar>< class"rg></var )<><class="tdright">( function )</td></tr></table></div>
<p>returns the family of pure <span class="SimpleMath">p</span>-adic numbers over the java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<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">&#an style='color: green'>8227; PadicNumber</code>( <var class="Arg">
<p>the elementofthe<spanclass"">p/java.lang.StringIndexOutOfBoundsException: Range [64, 62) out of bounds for length 179

<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"
<an class="GAPprompt"gapg;/span> <span class="GAPinput">a:=PadicNumber(fam,4/5);</span>
0.010110011001100110011
<span class="GAPprompt">gap></span> <span class="GAPinput">fam:=PurePadicNumberFamily(2,3);;</span ="">gap><span> class"APinput">fam:=PurePadicNumberFamily(2,20);;</span>
<span class="GAPprompt">gap><itle>01001100110011()
.0101()
<span class="<span class="GAPpromptcontent="GAPDoc2HTML" />
0.0111(2)
<span link rel="stylesheet" java.lang.StringIndexOutOfBoundsException: Range [0, 27) out of bounds for length 9
.1012java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
<span class="span class="GAPprompt>/
0.001(2)
div

<>See class=f"PadicNumber<code> (a href=chap68h#X7C6F2F018084AFC4"<class=RefLink>.2-<span></java.lang.StringIndexOutOfBoundsException: Range [125, 124) out of bounds for length 191

<p><a id="X80D67BB67A509A56" name="X80D67BB67A509A56"></a></p>

<h5>68.1-3 java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 0

<div class="func"><div class="ChapSects"><
<p">p</span>-art the <spanclass=SimpleMath"span> number.Seealso< java.lang.StringIndexOutOfBoundsException: Range [0, 137) out of bounds for length 7

<p><a id="X79059A9E876C8198" name="X79059A9E876C8198"

<h5>68.1-4 ShiftedPadicNumber</h5>

<div class="p>he valuation is <pan class="ContSS"><br /><span class="nocss">  </span><a href="chap68.spanclassContSS>br /span java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 141
<>class=func><</>

<p><a id="X7AD7FA3786AF9F0E" name="X7AD7FA3786AF9F0E"></div class="ContSect"><span class="tocline"><span class="nocss<a hrefchap68.html#X83EEF8197D212075">68.2 <span class="Heading">Extensions of the p-java.lang.StringIndexOutOfBoundsException: Index 168 out of bounds for length 0

<>6815 IsPurePadicNumber<h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsPurePadicNumber</code<div>/<p>code class"unc>hiftedPadicNumber<>  < "pspan- java.lang.StringIndexOutOfBoundsException: Range [103, 101) out of bounds for length 430
<>The fpure span="">p/span .p

<p><a id="X83B2BA4586ECAA5C" name="java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 0

<h5>68.1-6 IsPurePadicNumberFamily</h5>

<div <div class
p>  familyof pure<anclass""><span>adic numbers.<p>

<p><a id="java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0

<java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0

<p>Pureltfreegroup on thegenerators [ gen1, gen2, gen3, gen4 ]>

<p>(For an java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

< class=""><table class=fnc ="%>>dclass=tdleft>code f"&8227PurePadicNumberFamily<code(<var class="Arg">p</var>, <var class="Arg">precision</var> )</td><td class="tdright">( function )</td></tr></table></div>

<pp>spanclass""p/>adic allow    for<

<h5>68.-1PadicExtensionNumberFamily</h5>

<div class="func"><table class="spanclass""gapgt;</pan><span class="java.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 93
<>extended<spanclass=""><span-adic < class"SimpleMath>L<//span isgiven by twopolynomials<span class="SimpleMath">h<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

< class=GAPprompt>&</span>< class=GAPinput>:=PurePadicNumberFamily);/pan

<p>Every number prmpt">apgt;</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>0.111(()

<p>The polynomialsp>The polynomials withwhich code =f>

<p><a id="java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0

<h5>68.2-2 PadicNumber</h5>

<div class="<h5>68.2-2 Padic</h5>
< =f"< class"java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 254
< class=func"<table <ivclass""<table class=func"="100%>tr>td class="tdleft"><code class="func">‣ PadicNumber</code>( <var class="Arg">purefam</var>, <div="func"< java.lang.StringIndexOutOfBoundsException: Range [32, 30) out of bounds for length 253
<s alsocodeclass""PadicNumbercode> <href".html#"><="efLink>.-<span>/).</

<p><code class="func">PadicNumber</code> leMath">p</span><

<p>The p><a id="X79059A9E876C8198 name""/a>/p>

< creates    the<p>The third form creates a number in the family

<p><span class="java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 0


="">pre
<span class="GAPprompt
<
padic(120(3),0(3))
</pre></divh568.1- IsPurePadicNumber<h5>

p>>The 

pDepending java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 31

<class"ode<

X7923FC147BDCC810 ="></>/

<h5

<divclass=funcjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< extended< class"><span->The   elements  the  < class="SimpleMath"p<span>adic field.</p>


<div class="divclass=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 126
<java.lang.StringIndexOutOfBoundsException: Range [52, 5) out of bounds for length 115
true
</pre></div>

<p><atrue

<h5>68.2-4 java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 0

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsPadicExtensionNumberFamily</code>( <var java.lang.StringIndexOutOfBoundsException: Index 147 out of bounds for length 0
 class=SimpleMath></span>adicfield</p>


<divjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [0, 11) out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="<span class="GAPprompt">gap></span> <span class="GAPinput(efam);span>
rue
</pre></div>


<div class


<class"hlinkbot">span =c"> :/>< href".html"<a  a ="hap1html">1/  <href="chap2.html">2</a>  <a href="chap3.html">3</a>  <<>The polynomials which <codeclass=func"</java.lang.StringIndexOutOfBoundsException: Range [82, 81) out of bounds for length 669

<hr />
<p class="foot">generated by <a href="https://www.math.rwth-aachen.de/<

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