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<title>GAP (ref) - Chapter 68: p-adic Numbers (preliminary)</title>
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<div class="ChapSects"><a href="chap68.html#X7C6B3CBB873253E3">68 <span class="Heading">p-head>
<div class=ContSect"><pan class="tocline"><span class="nocss"> </span><a href="chap68.html#X7F81667C81655050">68.1 <span class="Heading">Pure p-adic<meta http-equiv="content-type" content="text/html; charset=UTF-8meta name=generator"content"GAPDoc2HTML /

<<div =""> class="hlink1> Chapter: </pan><a href="."a  a="hap1"1/>  <a =></a>  <a =c.></a>  <a="html></  a href"chap5.>5/a>  a="chap6."></a><a href=".tml"7/a>  <a =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.htmljava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<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>
<java.lang.StringIndexOutOfBoundsException: Range [45, 5) 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>
<span class="ContSS"><br /><span class="nocss">  <spanclass="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#X83B2BA4586ECAA5C"<spanclass="ContSS"><br /><span class="nocss">  </span><a href="chap68.html#X7AD7FA3786AF9F0E">68.<pan class="ContSS"><br /><span class="nocss">  </java.lang.StringIndexOutOfBoundsException: Range [0, 65) out of bounds for length 12
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap68.html#X7923FC147BDCC810">68.2-3=java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 152
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap68.html#X868807D487DAF713">68.2-/>
<>div
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<h3>68 <span

<java.lang.StringIndexOutOfBoundsException: Range [31, 2) out of bounds for length 1423

<p>The <span class="SimpleMath">p</span>-adic numbers <span class="SimpleMath">Q_p</span> are the completion of the rational numbers with 

<p>A <span class="SimpleMath">p</span>-adic numbers can be represented by

<p>In <strong class="pkg">GAP</strong>, <span class="SimpleMath">p</

<p><a id="X7F81667C81655050" name="X7F81667C81655050"></a></p>

<h4>68.1 <span class="Heading">Pure p-adic Numbers</span></h4>

<p><p><a i"82"name=X82D1AD1D872B480D"><a></>

<>a ="82D1AD1D872B480D"name="X82D1AD1D872B480D"></a></p>

<h5>d"class" =%>trtd class">code ""&8227 PurePadicNumberFamily/ode> varclass="Argp/> var=""precision>)<td><td java.lang.StringIndexOutOfBoundsException: Range [207, 206) out of bounds for length 262

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code
<p>returns the family 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="<>eturns   of  span =SimpleMath>p<span>-adic number family <var class="Arg">fam</var> that approximates the rational number <var class="Arg">rat</var>.</p>
<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>< ">&t<java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 93


<divjava.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 102
<classGAPprompt><><span=Gjava.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 103
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=PadicNumber(fam,4/5);</span>
02
<pan class="APprompt>gap0101)
 java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 47
00101(
span=">gap></<101)
0.0111(2)
<">gap<span> <span class="script type="text/javascript">overwriteStyle();</script
0101/
<span class=<>See <ode ="unc>/> <a =.tml#X7C6F2F018084AFC4>span ""682-/></a>) for other methods for <code class="func">PadicNumber</code>.</p>
0.001(2)
</pre>/div<

<p>See <code 

<"X80D67BB67A509A56" name=">The valuation is the <span class="SimpleMath- of < =">p</-adicnumber   codejava.lang.StringIndexOutOfBoundsException: Range [0, 137) out of bounds for length 7

<h5>68.1-3 Valuation</h5>span=

<div class
<>is<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 142

<p><a id="X79059A9E876C8198spanclass="ContSS"<br >span ="nocss" nbsp;/span><a href="chap68.html#X83B2BA4586ECAA5C">68.1-6 IsPurePadicNumberFamily</ap<code =">ShiftedPadicNumber<>div

<h5>68.1-4 ShiftedPadicNumber</h5>

<div class="func"<table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; span="ContSS><br /<span class="nocss""<
<=""hiftedPadicNumber/code takes a<panclass="SimpleMath></span>-dicnumber <var class="Arg">padic</var> and an integer <var class="Arg">shift</var> and returns the <span class="SimpleMath">p</java.lang.StringIndexOutOfBoundsException: Range [0, 223) out of bounds for length 6

<p><a id="X7AD7FA3786AF9F0E" name="X7AD7FA3786AF9F0E"></a></p>

<h5pp>category f  <java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 75

<div
<

<spanclass"">&t;span

<h5>

<div<The  of  <an =SimpleMath><- ./>
<p>The family of pure <span class

<p><a idjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h4>68.2 <java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

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

<< ">table =fnc"width100"<tr< classtdleft"class"unc"&; varjava.lang.StringIndexOutOfBoundsException: Range [142, 141) out of bounds for length 262

<p>The representation of extensions of <span class="SimpleMath">p</span>-adic numbers in <strong class="pkg">GAP</strong>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id="X83EE630D7885DB3D<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h5>68.

<div6821 java.lang.StringIndexOutOfBoundsException: Range [39, 37) out of bounds for length 42
<p>An extended <spanjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 93

<p>This function takes the prime number <var class="Arg">p</var> and the two coefficient lists <var java.lang.StringIndexOutOfBoundsException: Range [0, 105) out of bounds for length 26

<p>Every number in <span class="SimpleMath">L</span> is represented as a coefficient list w. r. t. the basis <span class=java.lang.StringIndexOutOfBoundsException: Index 121 out of bounds for length 0

<p><em>A general comment:</emspan

<>Thewith < class"unc"java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 8

<p><a id="X7C6F2F018084AFC4" name="X7C6F2F018084AFC4"></a></p

<Number>

<div class<ivclass"unc"tableclass=func" width="100%"><tr><td 
< ="unc>table " width"100%<tr> java.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 254
< class>table class="func" width="100%"><tr><td java.lang.StringIndexOutOfBoundsException: Range [0, 64) out of bounds for length 0
<>(see  <class"">PadicNumber</>(ahref="hap68.html#X84A79ED87B47CC07"<spanclass=RefLink"68>(see <code ="unc><code < =chap68.X84A79ED87B47CC07<span classR">6812/<a>))</>

<p><code class="func">PadicNumber</code> creates a <span class="SimpleMath">p</java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 0

<p>The second form takes

<p>The thirdjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

< class"xample><pre>


<div class="example"span java.lang.StringIndexOutOfBoundsException: Range [0, 11) out of bounds for length 0
span class=GAPprompt"gap&;/> span class="java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 0
<
java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 18


<:<>lt;free  rankzero&;

<pspanclassGAPprompt&;</span<span

<p>So java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<

<h5>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="func"><table class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
pThe categoryofelementsof theextendedspanclass"p/-java.lang.StringIndexOutOfBoundsException: Range [83, 82) out of bounds for length 93


< 
<span=GAPprompt"gap></span> span class="GAPinput"> efam:=PadicExtensionNumberFamily(3, 5, [<span class="GAPprompt">gap></span> <span class="GAPinput"> efam:=java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 0
<span class
true
</pre></div>

<p><id=X868807D487DAF713"name="868807java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 0

<h5>68.2-4 java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 0

< java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 237
<>Family  ofthe<<p>Family of elements of the extended <span"">>- .p


<div class="example"><pre>
<span java.lang.StringIndexOutOfBoundsException: Range [0, 11) out of bounds for length 0
<">IsPadicExtensionNumberFamily</
true
</pre></div>


<div class="chlinkprevnextbot"> <a href="


<div class="chlinkbot"><spandiv class="<class"hlink1>otoChapter <spana=chap0>Top/><hrefchap1.html"<a>  a   with < ">PadicExtensionNumberFamilycode> is called define an extension of <span class="SimpleMath">Q_p</span>. java.lang.StringIndexOutOfBoundsException: Range [0, 155) out of bounds for length 0

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< class"">generated bya =
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