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<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   non 2 <a href="ring_3.html#V3" title="RING_3:attr.3">-characteristic</a>   <a href="field_3.html#V3" title="FIELD_3:attr.3">polynomial_disjoint</a>   non  <a href="field_9.html#V3" title="FIELD_9:attr.3">quadratic_complete</a>  <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   non  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> holds   <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Olive" title="b1">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) is    <a href="field_8.html#M1" title="FIELD_8:mode.1">SplittingField</a> of  <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Olive" title="b1">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font> be   non  <a href="realalg1.html#NV8" title="REALALG1:NV.8">square</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) is    <a href="field_8.html#M1" title="FIELD_8:mode.1">SplittingField</a> of  <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">E</font> =  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">p</font> =  <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font> as    <a href="ring_4.html#M1" title="RING_4:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c1">F</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">b</font> =  <a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#D5" onmouseover="rs('field_7/D5')" onmouseout="rh()">FIELD_7:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">q</font> = <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c5">b</font>)</span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</font></span>)</span>)</span>)</span> as    <a href="ring_4.html#M1" title="RING_4:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p3"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#D10" onmouseover="rs('polynom3/D10')" onmouseout="rh()">POLYNOM3:def 10</a></span>;<br/>
<a NAME="E4:125"/>
 <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) is    <a href="c0sp1.html#M1" title="C0SP1:mode.1">Subring</a> of  <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <span class="p2">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D1" onmouseover="rs('field_4/D1')" onmouseout="rh()">FIELD_4:def 1</a></span>;<br/>
<a NAME="E5:125"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">H</font></span>: 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T17" onmouseover="rs('field_6/T17')" onmouseout="rh()">FIELD_6:17</a></span>;<br/>
<a NAME="E6:125"/><span class="lab"><font color="Green" title="E59">I</font></span>: 
 <a href="field_9.html#K2" title="FIELD_9:func.2">X-</a> <font color="Maroon" title="c5">b</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="field_9.html#K2" title="FIELD_9:func.2">X-</a> <span class="p1">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T21" onmouseover="rs('field_4/T21')" onmouseout="rh()">FIELD_4:21</a></span>;<br/>
<a NAME="E7:125"/><span class="lab"><font color="Green" title="E60">K</font></span>: 
 <a href="field_9.html#K3" title="FIELD_9:func.3">X+</a> <font color="Maroon" title="c5">b</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="field_9.html#K3" title="FIELD_9:func.3">X+</a> <span class="p1">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:125"><span class="lab"><font color="Green" title="E58">H</font></span></a>, <a class="ref" href="field_4.html#T21" onmouseover="rs('field_4/T21')" onmouseout="rh()">FIELD_4:21</a></span>;<br/>
<span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c5">b</font>)</span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</font></span>)</span>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="field_9.html#K2" title="FIELD_9:func.2">X-</a> <span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p1">(<span class="default"><a href="field_9.html#K3" title="FIELD_9:func.3">X+</a> <span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:125"><span class="lab"><font color="Green" title="E59">I</font></span></a>, <a class="txt" href="#E7:125"><span class="lab"><font color="Green" title="E60">K</font></span></a>, <a class="ref" href="field_4.html#T17" onmouseover="rs('field_4/T17')" onmouseout="rh()">FIELD_4:17</a></span>
<br/>.= 
 <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#T56" target="_self" onmouseover="rs('field_9/T56')" onmouseout="rh()">Fi1a</a></span>
;<br/>
<a NAME="E9:125"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">D</font></span>: 
 <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p3"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span></span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <font color="Maroon" title="c6">q</font>
 
;<br/>
<a NAME="E10:125"/>
(  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c5">b</font>) is    <a href="ring_5.html#M1" title="RING_5:mode.1">Ppoly</a> of  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) &amp;  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</font></span>)</span>) is    <a href="ring_5.html#M1" title="RING_5:mode.1">Ppoly</a> of  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ring_5.html#T51" onmouseover="rs('ring_5/T51')" onmouseout="rh()">RING_5:51</a></span>;<br/>
<a NAME="E11:125"/><span class="kw">then </span>
<font color="Maroon" title="c6">q</font> is    <a href="ring_5.html#M1" title="RING_5:mode.1">Ppoly</a> of  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ring_5.html#T52" onmouseover="rs('ring_5/T52')" onmouseout="rh()">RING_5:52</a></span>;<br/>
<a NAME="E12:125"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A</font></span>: 
 <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a>  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:125"><span class="lab"><font color="Green" title="E61">D</font></span></a>, <a class="ref" href="field_4.html#D5" onmouseover="rs('field_4/D5')" onmouseout="rh()">FIELD_4:def 5</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="125_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">U</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font>  st  <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Olive" title="b1">U</font> &amp; <font color="Olive" title="b1">U</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) holds <br/> <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a> <font color="Olive" title="b1">U</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">U</font> be   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Maroon" title="c7">U</font> &amp; <font color="Maroon" title="c7">U</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) implies  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a> <font color="Maroon" title="c7">U</font> )</span><br/><span class="kw">assume </span><a NAME="E1:125_2"/><span class="lab"><font color="Green" title="E63">D0</font></span>: 
(  <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Maroon" title="c7">U</font> &amp; <font color="Maroon" title="c7">U</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a> <font color="Maroon" title="c7">U</font></span><br/><a NAME="E2:125_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">D3</font></span>: 
 <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) is <font color="Maroon" title="c7">U</font> <a href="field_4.html#V1" title="FIELD_4:attr.1">-extending</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T7" onmouseover="rs('field_4/T7')" onmouseout="rh()">FIELD_4:7</a></span>;<br/><a NAME="E3:125_2"/><span class="lab"><font color="Green" title="E65">D4</font></span>: 
 <a href="field_4.html#K1" title="FIELD_4:func.1">Roots</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<span class="p1">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:125_2"><span class="lab"><font color="Green" title="E64">D3</font></span></a>, <a class="txt" href="#E1:125_2"><span class="lab"><font color="Green" title="E63">D0</font></span></a>, <a class="txt" href="#E12:125"><span class="lab"><font color="Green" title="E62">A</font></span></a>, <a class="ref" href="field_8.html#T27" onmouseover="rs('field_8/T27')" onmouseout="rh()">FIELD_8:27</a></span>;<br/><a NAME="E4:125_2"/>
 <a href="field_4.html#K1" title="FIELD_4:func.1">Roots</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c4">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="field_7.html#K1" title="FIELD_7:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span><a href="field_7.html#K1" title="FIELD_7:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#T57" target="_self" onmouseover="rs('field_9/T57')" onmouseout="rh()">Fi2a</a></span>;<br/><a NAME="E5:125_2"/><span class="kw">then </span>
 <a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="field_4.html#K1" title="FIELD_4:func.1">Roots</a> (<span class="p1">(<span class="default"><a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p2"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c4">p</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><a NAME="E6:125_2"/><span class="kw">then </span>
 <a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:125_2"><span class="lab"><font color="Green" title="E65">D4</font></span></a></span>;<br/><a NAME="E7:125_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">D1</font></span>: 
<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">U</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E8:125_2"/><span class="lab"><font color="Green" title="E67">D2</font></span>: 
<font color="Maroon" title="c7">U</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of  <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <span class="p2">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:125_2"><span class="lab"><font color="Green" title="E63">D0</font></span></a>, <a class="ref" href="ec_pf_1.html#T5" onmouseover="rs('ec_pf_1/T5')" onmouseout="rh()">EC_PF_1:5</a></span>;<br/><a NAME="E9:125_2"/>
<font color="Maroon" title="c1">F</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of <font color="Maroon" title="c7">U</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T7" onmouseover="rs('field_4/T7')" onmouseout="rh()">FIELD_4:7</a></span>;<br/><a NAME="E10:125_2"/><span class="kw">then </span>
 <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of <font color="Maroon" title="c7">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:125_2"><span class="lab"><font color="Green" title="E66">D1</font></span></a>, <a class="txt" href="#E8:125_2"><span class="lab"><font color="Green" title="E67">D2</font></span></a>, <a class="ref" href="field_6.html#T37" onmouseover="rs('field_6/T37')" onmouseout="rh()">FIELD_6:37</a></span>;<br/><span class="kw">hence </span><a NAME="E11:125_2"/>
 <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) <a href="field_7.html#R1" title="FIELD_7:pred.1">==</a> <font color="Maroon" title="c7">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:125_2"><span class="lab"><font color="Green" title="E63">D0</font></span></a>, <a class="ref" href="field_7.html#D2" onmouseover="rs('field_7/D2')" onmouseout="rh()">FIELD_7:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E14:125"/>
 <a href="field_6.html#K6" title="FIELD_6:func.6">FAdj</a> (<font color="Maroon" title="c1">F</font>,<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="field_9.html#K16" title="FIELD_9:func.16">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) is    <a href="field_8.html#M1" title="FIELD_8:mode.1">SplittingField</a> of  <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:125"><span class="lab"><font color="Green" title="E62">A</font></span></a>, <a class="ref" href="field_8.html#D1" onmouseover="rs('field_8/D1')" onmouseout="rh()">FIELD_8:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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