<?xml version="1.0"?>
<div class="add">

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="90_4_1_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">  not  <a href="field_9.html#K11" title="FIELD_9:func.11">X^2+X+1</a>  is  <a href="ring_2.html#NV6" title="RING_2:NV.6">reducible</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:90_4_1_1"/>
 <a href="field_9.html#K11" title="FIELD_9:func.11">X^2+X+1</a>  is  <a href="ring_2.html#NV6" title="RING_2:NV.6">reducible</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"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c1">q</font> being    <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="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:90_4_1_1"/><span class="lab"><font color="Green" title="E45">C2</font></span>: 
( <font color="Maroon" title="c1">q</font> <a href="ring_4.html#R1" title="RING_4:pred.1">divides</a>  <a href="field_9.html#K11" title="FIELD_9:func.11">X^2+X+1</a>  &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c1">q</font> &amp;  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c1">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <a href="field_9.html#K11" title="FIELD_9:func.11">X^2+X+1</a> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ring_4.html#T41" onmouseover="rs('ring_4/T41')" onmouseout="rh()">RING_4:41</a></span>;<br/><a NAME="E4:90_4_1_1"/>
 <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <a href="field_9.html#K11" title="FIELD_9:func.11">X^2+X+1</a> <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#T18" target="_self" onmouseover="rs('field_9/T18')" onmouseout="rh()">qua4</a></span>;<br/><a NAME="E5:90_4_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c1">q</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_4_1_1"><span class="lab"><font color="Green" title="E45">C2</font></span></a>, <a class="ref" href="int_1.html#T7" onmouseover="rs('int_1/T7')" onmouseout="rh()">INT_1:7</a></span>;<br/><a NAME="E6:90_4_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c1">q</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E8:90_4_1_1"/><span class="lab"><font color="Green" title="E46">C3</font></span>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span> &amp; <font color="Maroon" title="c1">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">x</font> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c3">a</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_4_1_1"><span class="lab"><font color="Green" title="E45">C2</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a>, <a class="ref" href="hurwitz.html#T28" onmouseover="rs('hurwitz/T28')" onmouseout="rh()">HURWITZ:28</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c4">p</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <span class="p1">(<span class="default"><a href="ring_3.html#NK1" title="RING_3:NK.1">Z/</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E10:90_4_1_1"/><span class="lab"><font color="Green" title="E47">C4</font></span>: 
 <a href="field_9.html#K11" title="FIELD_9:func.11">X^2+X+1</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c1">q</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c4">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_4_1_1"><span class="lab"><font color="Green" title="E45">C2</font></span></a>, <a class="ref" href="ring_4.html#T1" onmouseover="rs('ring_4/T1')" onmouseout="rh()">RING_4:1</a></span>;<br/><span class="kw">thus </span><a NAME="E11:90_4_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:90_4_1_1"><span class="lab"><font color="Green" title="E46">C3</font></span></a>, <a class="txt" href="#E10:90_4_1_1"><span class="lab"><font color="Green" title="E47">C4</font></span></a>, <a class="ref" href="field_9.html#T42" target="_self" onmouseover="rs('field_9/T42')" onmouseout="rh()">z24</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="E2:90_4_1"/>
 <a href="field_9.html#K11" title="FIELD_9:func.11">X^2+X+1</a>  is  <a href="ring_2.html#V5" title="RING_2:attr.5">irreducible</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"> verum</span><br/>


</div>
