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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="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">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font><br/>  for <font color="Olive" title="b2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font>  st (  for <font color="Olive" title="b3">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font>  holds  not <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a>  ) holds <br/> for <font color="Olive" title="b3">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">E</font>  st <font color="Olive" title="b3">b</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> 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"><font color="Olive" title="b3">b</font></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="b2">a</font> &amp;  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</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"><font color="Olive" title="b3">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">E</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">  for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font>  st (  for <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font>  holds  not <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a>  ) holds <br/> for <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font>  st <font color="Olive" title="b2">b</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> 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"><font color="Olive" title="b2">b</font></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> &amp;  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</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"><font color="Olive" title="b2">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <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"> ( (  for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font>  holds  not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a>  ) implies  for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font>  st <font color="Olive" title="b1">b</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> 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"><font color="Olive" title="b1">b</font></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="c3">a</font> &amp;  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</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"><font color="Olive" title="b1">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:152"/><span class="lab"><font color="Green" title="E68">AS1</font></span>: 
 for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font>  holds  not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</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">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font>  st <font color="Olive" title="b1">b</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> 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"><font color="Olive" title="b1">b</font></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="c3">a</font> &amp;  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</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"><font color="Olive" title="b1">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</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"> ( <font color="Maroon" title="c4">b</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> 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"><font color="Maroon" title="c4">b</font></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="c3">a</font> &amp;  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</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"><font color="Maroon" title="c4">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:152"/><span class="lab"><font color="Green" title="E69">AS2</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="o_ring_1.html#K1" title="O_RING_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</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"><font color="Maroon" title="c4">b</font></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="c3">a</font> &amp;  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</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"><font color="Maroon" title="c4">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">a1</font> = <font color="Maroon" title="c3">a</font> as   <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="c2">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:152"><span class="lab"><font color="Green" title="E69">AS2</font></span></a></span>;<br/>
<a NAME="E4:152"/>
<font color="Maroon" title="c1">F</font> is    <a href="c0sp1.html#M1" title="C0SP1:mode.1">Subring</a> of <font color="Maroon" title="c2">E</font>
 
<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="E5:152"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">H1</font></span>: 
(  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font> &amp;  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font> &amp;  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c3">a</font> )
 
<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>, <a class="ref" href="c0sp1.html#D3" onmouseover="rs('c0sp1/D3')" onmouseout="rh()">C0SP1:def 3</a></span>;<br/>
<span class="lab"><font color="Green" title="E71">H0</font></span>: <font color="Maroon" title="c4">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">b</font></span>)</span> = 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T8" onmouseover="rs('vectsp_1/T8')" onmouseout="rh()">VECTSP_1:8</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c5">a1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:152"><span class="lab"><font color="Green" title="E69">AS2</font></span></a>, <a class="ref" href="o_ring_1.html#D1" onmouseover="rs('o_ring_1/D1')" onmouseout="rh()">O_RING_1:def 1</a></span>
;<br/>
<span class="lab"><font color="Green" title="E72">H2</font></span>:  <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c3">a</font> = 
<span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</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">a1</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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">E</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:152"><span class="lab"><font color="Green" title="E70">H1</font></span></a></span>
<br/>.= 
<span class="p1"><a href="field_9.html#K4" title="FIELD_9:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">b</font></span>)</span></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"><font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">E</font></span>)</span></span><a href="field_9.html#K4" title="FIELD_9:func.4">%&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:152"><span class="lab"><font color="Green" title="E71">H0</font></span></a>, <a class="ref" href="rlvect_1.html#T5" onmouseover="rs('rlvect_1/T5')" onmouseout="rh()">RLVECT_1:5</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="c4">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="c4">b</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="field_9.html#E38"><span class="lab"><font color="Green" title="E27">lemred3z</font></span></a></span>
;<br/>
<a NAME="E8:152"/>
(  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>) is    <a href="ring_5.html#M1" title="RING_5:mode.1">Ppoly</a> of <font color="Maroon" title="c2">E</font> &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="c4">b</font></span>)</span>) is    <a href="ring_5.html#M1" title="RING_5:mode.1">Ppoly</a> of <font color="Maroon" title="c2">E</font> )
 
<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/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">p</font> =  <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c3">a</font> as    <a href="ring_5.html#M1" title="RING_5:mode.1">Ppoly</a> of <font color="Maroon" title="c2">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:152"><span class="lab"><font color="Green" title="E72">H2</font></span></a>, <a class="ref" href="ring_5.html#T52" onmouseover="rs('ring_5/T52')" onmouseout="rh()">RING_5:52</a></span>;<br/>
<a NAME="E10:152"/><span class="lab"><font color="Green" title="E73">H3</font></span>: 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <font color="Maroon" title="c6">p</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c3">a</font>
 
;<br/>
<a NAME="E11:152"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">H11</font></span>: 
 <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c3">a</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Maroon" title="c2">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D5" onmouseover="rs('field_4/D5')" onmouseout="rh()">FIELD_4:def 5</a></span>;<br/>
<a NAME="E12:152"/><span class="lab"><font color="Green" title="E75">H5</font></span>: 
(  <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="c4">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="c4">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="c4">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="c4">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="lab"><font color="Green" title="E76">H9</font></span>:  <a href="field_4.html#K1" title="FIELD_4:func.1">Roots</a> (<font color="Maroon" title="c2">E</font>,<span class="p1">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c3">a</font></span>)</span>) = 
 <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <span class="p1">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c5">a1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:152"><span class="lab"><font color="Green" title="E70">H1</font></span></a>, <a class="ref" href="field_7.html#T13" onmouseover="rs('field_7/T13')" onmouseout="rh()">FIELD_7:13</a></span>
<br/>.= 
<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">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="c4">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="txt" href="#E7:152"><span class="lab"><font color="Green" title="E72">H2</font></span></a>, <a class="txt" href="#E5:152"><span class="lab"><font color="Green" title="E70">H1</font></span></a>, <a class="txt" href="#E12:152"><span class="lab"><font color="Green" title="E75">H5</font></span></a>, <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="c4">b</font>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">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="E14:152"/><span class="kw">then </span>
 <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c3">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="field_7.html#K1" title="FIELD_7:func.1">{</a><span class="default"><font color="Maroon" title="c4">b</font>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">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="txt" href="#E10:152"><span class="lab"><font color="Green" title="E73">H3</font></span></a>, <a class="ref" href="field_8.html#T30" onmouseover="rs('field_8/T30')" onmouseout="rh()">FIELD_8:30</a>, <a class="ref" href="field_4.html#D5" onmouseover="rs('field_4/D5')" onmouseout="rh()">FIELD_4:def 5</a></span>;<br/>
<a NAME="E15:152"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">H</font></span>: 
 <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c3">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"><font color="Maroon" title="c4">b</font></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="field_9.html#T9" target="_self" onmouseover="rs('field_9/T9')" onmouseout="rh()">ext1</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="152_4"><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="c3">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"><font color="Maroon" title="c4">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) holds <br/><font color="Olive" title="b1">U</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</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"><font color="Maroon" title="c4">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</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="c3">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"><font color="Maroon" title="c4">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) implies <font color="Maroon" title="c7">U</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</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"><font color="Maroon" title="c4">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) )</span><br/><span class="kw">assume </span><a NAME="E1:152_4"/><span class="lab"><font color="Green" title="E78">AS</font></span>: 
(  <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c3">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"><font color="Maroon" title="c4">b</font></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"> <font color="Maroon" title="c7">U</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</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"><font color="Maroon" title="c4">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span><br/><a NAME="E2:152_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">H8</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"><font color="Maroon" title="c4">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>) is   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c7">U</font> &amp; <font color="Maroon" title="c2">E</font> is   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</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"><font color="Maroon" title="c4">b</font></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="field_4.html#T7" onmouseover="rs('field_4/T7')" onmouseout="rh()">FIELD_4:7</a></span>;<br/><a NAME="E3:152_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">H6</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 <font color="Maroon" title="c2">E</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="E4:152_4"/><span class="lab"><font color="Green" title="E81">H7</font></span>: 
<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="E5:152_4"/>
 <a href="field_4.html#K1" title="FIELD_4:func.1">Roots</a> (<font color="Maroon" title="c2">E</font>,<span class="p1">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c3">a</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="field_4.html#K1" title="FIELD_4:func.1">Roots</a> (<font color="Maroon" title="c7">U</font>,<span class="p1">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c3">a</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:152"><span class="lab"><font color="Green" title="E74">H11</font></span></a>, <a class="txt" href="#E1:152_4"><span class="lab"><font color="Green" title="E78">AS</font></span></a>, <a class="txt" href="#E2:152_4"><span class="lab"><font color="Green" title="E79">H8</font></span></a>, <a class="ref" href="field_8.html#T29" onmouseover="rs('field_8/T29')" onmouseout="rh()">FIELD_8:29</a></span>;<br/><a NAME="E6:152_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">H10</font></span>: 
<font color="Maroon" title="c4">b</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> (<font color="Maroon" title="c7">U</font>,<span class="p1">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c3">a</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:152"><span class="lab"><font color="Green" title="E76">H9</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="152_4_1"><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">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> holds <br/><font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN: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></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( <font color="Maroon" title="c8">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> implies <font color="Maroon" title="c8">o</font> <a href="hidden.html#R2" title="HIDDEN: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><br/><span class="kw">assume </span><a NAME="E1:152_4_1"/>
<font color="Maroon" title="c8">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">b</font></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"> <font color="Maroon" title="c8">o</font> <a href="hidden.html#R2" title="HIDDEN: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><br/><a NAME="E2:152_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</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/><span class="kw">hence </span><a NAME="E3:152_4_1"/>
<font color="Maroon" title="c8">o</font> <a href="hidden.html#R2" title="HIDDEN: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="#E6:152_4"><span class="lab"><font color="Green" title="E82">H10</font></span></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><a NAME="E8:152_4"/><span class="kw">then </span>
<span class="p1"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c4">b</font></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>
 ;<br/><a NAME="E9:152_4"/><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"><font color="Maroon" title="c4">b</font></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="#E3:152_4"><span class="lab"><font color="Green" title="E80">H6</font></span></a>, <a class="txt" href="#E4:152_4"><span class="lab"><font color="Green" title="E81">H7</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="E10:152_4"/>
<font color="Maroon" title="c7">U</font> <a href="field_7.html#R1" title="FIELD_7:pred.1">==</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"><font color="Maroon" title="c4">b</font></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="#E1:152_4"><span class="lab"><font color="Green" title="E78">AS</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="E17:152"/>
 <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"><font color="Maroon" title="c4">b</font></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="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:152"><span class="lab"><font color="Green" title="E77">H</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">  <a href="field_4.html#K3" title="FIELD_4:func.3">deg</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"><font color="Maroon" title="c4">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span><br/>

<a NAME="E18:152"/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c5">a1</font></span>)</span>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:152"><span class="lab"><font color="Green" title="E69">AS2</font></span></a>, <a class="ref" href="field_9.html#T70" target="_self" onmouseover="rs('field_9/T70')" onmouseout="rh()">m105</a></span>;<br/>
<a NAME="E19:152"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">H12</font></span>: 
 <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<span class="p1">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c3">a</font></span>)</span>,<font color="Maroon" title="c4">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:152"><span class="lab"><font color="Green" title="E70">H1</font></span></a>, <a class="ref" href="field_4.html#T26" onmouseover="rs('field_4/T26')" onmouseout="rh()">FIELD_4:26</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">b1</font> = <font color="Maroon" title="c4">b</font> as  <font color="Maroon" title="c1">F</font> <a href="field_6.html#V2" title="FIELD_6:attr.2">-algebraic</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="field_6.html#T43" onmouseover="rs('field_6/T43')" onmouseout="rh()">FIELD_6:43</a></span>;<br/>
<a NAME="E21:152"/>
 <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c3">a</font> is  <a href="ring_2.html#V5" title="RING_2:attr.5">irreducible</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#T33" target="_self" onmouseover="rs('field_9/T33')" onmouseout="rh()">naH2</a>, <a class="txt" href="#E1:152"><span class="lab"><font color="Green" title="E68">AS1</font></span></a>, <a class="ref" href="o_ring_1.html#D2" onmouseover="rs('o_ring_1/D2')" onmouseout="rh()">O_RING_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E79">H11</font></span>:  <a href="field_6.html#NK9" title="FIELD_6:NK.9">MinPoly</a> (<font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c1">F</font>) = 
 <a href="ring_4.html#K3" title="RING_4:func.3">NormPolynomial</a> <span class="p1">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c3">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:152"><span class="lab"><font color="Green" title="E78">H12</font></span></a>, <a class="ref" href="field_8.html#T15" onmouseover="rs('field_8/T15')" onmouseout="rh()">FIELD_8:15</a></span>
<br/>.= 
 <a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c3">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ring_4.html#T24" onmouseover="rs('ring_4/T24')" onmouseout="rh()">RING_4:24</a></span>
;<br/>
<a NAME="E23:152"/>
 <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <span class="p1">(<span class="default"><a href="field_9.html#K5" title="FIELD_9:func.5">X^2-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_9.html#D4" target="_self" onmouseover="rs('field_9/D4')" onmouseout="rh()">defquadr</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:152"/>
 <a href="field_4.html#K3" title="FIELD_4:func.3">deg</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"><font color="Maroon" title="c4">b</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span>)</span>)</span>,<font color="Maroon" title="c1">F</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:152"><span class="lab"><font color="Green" title="E79">H11</font></span></a>, <a class="ref" href="field_6.html#T67" onmouseover="rs('field_6/T67')" onmouseout="rh()">FIELD_6:67</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>
