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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_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="Olive" title="b1">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="Olive" title="b1">a</font></span>)</span>) <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="Olive" title="b1">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="Olive" title="b1">a</font></span>)</span></span>)</span></span><a href="field_7.html#K1" title="FIELD_7:func.1">}</a></span></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_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="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><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="E4:124"/><span class="lab"><font color="Green" title="E57">H</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    <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="E5:124"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">H1</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="E6:124"/>
(  <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c5">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> &amp;  <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</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#T18" onmouseover="rs('ring_5/T18')" onmouseout="rh()">RING_5:18</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E59">A</font></span>:  <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<span class="p3">(<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>)</span> = 
<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c5">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">b</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="uproots.html#T23" onmouseover="rs('uproots/T23')" onmouseout="rh()">UPROOTS:23</a></span>
<br/>.= 
<span class="p1"><a href="field_7.html#K1" title="FIELD_7:func.1">{</a><span class="default"><font color="Maroon" title="c5">b</font>,<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><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="enumset1.html#T1" onmouseover="rs('enumset1/T1')" onmouseout="rh()">ENUMSET1:1</a></span>
;<br/>
<a NAME="E8:124"/><span class="lab"><font color="Green" title="E60">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="E9:124"/><span class="lab"><font color="Green" title="E61">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="#E5:124"><span class="lab"><font color="Green" title="E58">H1</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="#E8:124"><span class="lab"><font color="Green" title="E60">I</font></span></a>, <a class="txt" href="#E9:124"><span class="lab"><font color="Green" title="E61">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="E11:124"/><span class="kw">then </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>,<font color="Maroon" title="c4">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <font color="Maroon" title="c6">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#T13" onmouseover="rs('field_7/T13')" onmouseout="rh()">FIELD_7:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:124"/>
 <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="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="txt" href="#E7:124"><span class="lab"><font color="Green" title="E59">A</font></span></a>, <a class="txt" href="#E4:124"><span class="lab"><font color="Green" title="E57">H</font></span></a>, <a class="ref" href="field_6.html#T17" onmouseover="rs('field_6/T17')" onmouseout="rh()">FIELD_6:17</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>
