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<div class="ChapSects"><a href="chap67.html#X85732CEF7ECFCA68">67 <span class="Heading">Algebraic extensions of fields</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap67.html#X7AD9B24E78ADC27F">67.1 <span class="Heading">Creation of Algebraic Extensions</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap67.html#X7CDA90537D2BAC8A">67.1-1 AlgebraicExtension</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap67.html#X811F10217F12B3F9">67.1-2 IsAlgebraicExtension</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap67.html#X819C7E6F78817F1E">67.2 <span class="Heading">Elements in Algebraic Extensions</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap67.html#X79695C887FD0AEAB">67.2-1 IsAlgebraicElement</a></span>
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<h3>67 <span class="Heading">Algebraic extensions of fields</span></h3>
<p>If we adjoin a root <span class="SimpleMath">α</span> of an irreducible polynomial <span class="SimpleMath">f ∈ K[x]</span> to the field <span class="SimpleMath">K</span> we get an <em>algebraic extension</em> <span class="SimpleMath">K(α)</span>, which is again a field. We call <span class="SimpleMath">K</span> the <em>base field</em> of <span class="SimpleMath">K(α)</span>.</p>
<p>By Kronecker's construction, we may identify <span class="SimpleMath">K(α)</span> with the factor ring <span class="SimpleMath">K[x]/(f)</span>, an identification that also provides a method for computing in these extension fields.</p>
<p>It is important to note that different extensions of the same field are entirely different (and its elements lie in different families), even if mathematically one could be embedded in the other one.</p>
<p>Currently <strong class="pkg">GAP</strong> only allows extension fields of fields <span class="SimpleMath">K</span>, when <span class="SimpleMath">K</span> itself is not an extension field.</p>
<p><a id="X7AD9B24E78ADC27F" name="X7AD9B24E78ADC27F"></a></p>
<h4>67.1 <span class="Heading">Creation of Algebraic Extensions</span></h4>
<p><a id="X7CDA90537D2BAC8A" name="X7CDA90537D2BAC8A"></a></p>
<h5>67.1-1 AlgebraicExtension</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlgebraicExtension</code>( <var class="Arg">K</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>constructs an extension <var class="Arg">L</var> of the field <var class="Arg">K</var> by one root of the irreducible polynomial <var class="Arg">f</var>, using Kronecker's construction. <var class="Arg">L</var> is a field whose <code class="func">LeftActingDomain</code> (<a href="chap57.html#X86F070E0807DC34E"><span class="RefLink">57.1-11</span></a>) value is <var class="Arg">K</var>. The polynomial <var class="Arg">f</var> is the <code class="func">DefiningPolynomial</code> (<a href="chap58.html#X7ADDCBF47E2ED3D4"><span class="RefLink">58.2-7</span></a>) value of <var class="Arg">L</var> and the attribute <code class="func">RootOfDefiningPolynomial</code> (<a href="chap58.html#X8173DA4982DB1E8A"><span class="RefLink">58.2-8</span></a>) of <var class="Arg">L</var> holds a root of <var class="Arg">f</var> in <var class="Arg">L</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=Indeterminate(Rationals,"x");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">p:=x^4+3*x^2+1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">e:=AlgebraicExtension(Rationals,p);</span>
<algebraic extension over the Rationals of degree 4>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsField(e);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=RootOfDefiningPolynomial(e);</span>
a
</pre></div>
<p><a id="X811F10217F12B3F9" name="X811F10217F12B3F9"></a></p>
<h5>67.1-2 IsAlgebraicExtension</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAlgebraicExtension</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>is the category of algebraic extensions of fields.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAlgebraicExtension(e);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAlgebraicExtension(Rationals);</span>
false
</pre></div>
<p><a id="X819C7E6F78817F1E" name="X819C7E6F78817F1E"></a></p>
<h4>67.2 <span class="Heading">Elements in Algebraic Extensions</span></h4>
<p>According to Kronecker's construction, the elements of an algebraic extension are considered to be polynomials in the primitive element. The elements of the base field are represented as polynomials of degree zero. <strong class="pkg">GAP</strong> therefore displays elements of an algebraic extension as polynomials in an indeterminate "a", which is a root of the defining polynomial of the extension. Polynomials of degree zero are displayed with a leading exclamation mark to indicate that they are different from elements of the base field.</p>
<p>The usual field operations are applicable to algebraic elements.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">a^3/(a^2+a+1);</span>
-1/2*a^3+1/2*a^2-1/2*a
<span class="GAPprompt">gap></span> <span class="GAPinput">a*(1/a);</span>
!1
</pre></div>
<p>The external representation of algebraic extension elements are the polynomial coefficients in the primitive element <code class="code">a</code>, the operations <code class="func">ExtRepOfObj</code> (<a href="chap79.html#X8542B32A8206118C"><span class="RefLink">79.16-1</span></a>) and <code class="func">ObjByExtRep</code> (<a href="chap79.html#X8542B32A8206118C"><span class="RefLink">79.16-1</span></a>) can be used for conversion.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ExtRepOfObj(One(a));</span>
[ 1, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ExtRepOfObj(a^3+2*a-9);</span>
[ -9, 2, 0, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ObjByExtRep(FamilyObj(a),[3,19,-27,433]);</span>
433*a^3-27*a^2+19*a+3
</pre></div>
<p><strong class="pkg">GAP</strong> does <em>not</em> embed the base field in its algebraic extensions and therefore lists which contain elements of the base field and of the extension are not homogeneous and thus cannot be used as polynomial coefficients or to form matrices. The remedy is to multiply the list(s) with the value of the attribute <code class="func">One</code> (<a href="chap31.html#X8046262384895B2A"><span class="RefLink">31.10-2</span></a>) of the extension which will embed all entries in the extension.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m:=[[1,a],[0,1]];</span>
[ [ 1, a ], [ 0, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMatrix(m);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">m:=m*One(e);</span>
[ [ !1, a ], [ !0, !1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMatrix(m);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">m^2;</span>
[ [ !1, 2*a ], [ !0, !1 ] ]
</pre></div>
<p><a id="X79695C887FD0AEAB" name="X79695C887FD0AEAB"></a></p>
<h5>67.2-1 IsAlgebraicElement</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAlgebraicElement</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>is the category for elements of an algebraic extension.</p>
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