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<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>

<p>  class"SimpleMathp<span>-adicnumbers./>

family<java.lang.StringIndexOutOfBoundsException: Range [56, 54) out of bounds for length 73

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<h4>68.1 <span class="Heading">

<><frefree

<p><a id="X82D1AD1D872B480D

<h5>68.1-1 PurePadicNumberFamily</h5>

iv="unctable " width"100tr><=< =unc"8227 PurePadicNumberFamily/> java.lang.StringIndexOutOfBoundsException: Range [142, 141) out of bounds for length 262
<p>returns the family of pure <span class="SimpleMath">p</span>-adic numbers over java.lang.StringIndexOutOfBoundsException: Index 1 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"
p

<p< =SimpleMath><span> numbers theusualoperations fields


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<h5>68.1-3 Valuation</h5>

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<h5>68.1-4 p>The third formanumber in java.lang.StringIndexOutOfBoundsException: Range [0, 48) out of bounds for length 0

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<p><a id="p><em>A word of warningDepending&java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 31

<h5>68.1-6 IsPurePadicNumberFamily=c">

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<h4>68.2 <span class="Heading">p>The category of elements of the <pan ="SimpleMath"p/>Thejava.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 93

<p>The class=GAPprompt>>/><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 126

<p>(For anspan class="GAPprompt">gap></span> <span class="GAPinput">IsPadicExtensionNumber(PadicNumber(efam,7/9));</span>

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<p><a id="X83EE630D7885DB3D

<h5>68.2-1 PadicExtensionNumberFamily</h5>

<div class
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<p>This function takes thejava.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 104

<p>Every number in <span class="SimpleMath">java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

<p

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<h5>68.2-2 PadicNumber</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class<
<div class="p=foot> <hrefjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 7
<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">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 &tfree grouponthe [,y]&;

<p>The second form takes aspan ="GAPprompt"gap></span> <span class="GAPinput">GeneratorsOfGroup(F){[1..4]};</span>

<p>The third form creates a number in java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 123

<p><span class="SimpleMath">p</span>-adic numbers admit the usual field operations.</p>


<div class="example"><pre>
<class">gt;</span> <span class="GAPinput">efam:=adicExtensionNumberFamily35 [11,][11);/pan>
<span class="GAPprompt">gap></span> <span class="GAPinput">PadicNumber(efam,7/9);</span>
padic(120(3),0<pre<div>
</pre></div>

<p><em>A word of warning:</em></p>

<p>Depending on the actual representation of quotients, precision may seem to <q>vanish</q>. For example in <code class="code">PadicExtensionNumberFamily(3, 5, [1,1,1]

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<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">&#an style='color: green'>8227; 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.</


<div class="example"><java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<pan =""ap></ spanclass=GAPinput>efam=3 ,[11] 11);/>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPadicExtensionNumber(PadicNumber(efam,pArg><var  agroup,whose generators   bysymbols (orexamplea group, a finitely presented group or a pc group) this function assigns these generators to global variables with the same names.</p>
true
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<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">&#an style='color: green'>8227; IsPadicExtensionNumberFamily</code>( <var class="Arg">fam</var> )</td><td class="tdright">( category )</td></tr></table></div>
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true
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