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<div class="ChapSects"><a href="chap68_mj.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_mj.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_mj.html#X82D1AD1D872B480D">68.1-1 PurePadicNumberFamily</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap68_mj.html#X84A79ED87B47CC07">68.1-2 PadicNumber</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap68_mj.html#X80D67BB67A509A56">68.1-3 Valuation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap68_mj.html#X79059A9E876C8198">68.1-4 ShiftedPadicNumber</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap68_mj.html#X7AD7FA3786AF9F0E">68.1-5 IsPurePadicNumber</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap68_mj.html#X83B2BA4586ECAA5C">68.1-6 IsPurePadicNumberFamily</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap68_mj.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_mj.html#X83EE630D7885DB3D">68.2-1 PadicExtensionNumberFamily</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap68_mj.html#X7C6F2F018084AFC4">68.2-2 PadicNumber</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap68_mj.html#X7923FC147BDCC810">68.2-3 IsPadicExtensionNumber</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap68_mj.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">\(\nu_p( p^v \cdot a / b) = v\)</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">( function )</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">( operation )</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_mj.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. See also <code class="func">PValuation</code> (<a href="chap15_mj.html#X8243EAA586D78ED4"><span class="RefLink">15.7-1</span></a>).</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_mj.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">\(\{ 1, x, x^2, \ldots, y, xy, x^2 y, \ldots \}\)</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_mj.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">\(\{ 1, \ldots, x^{{f-1}} \cdot y^{{e-1}} \}\)</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>
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