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	<title>Dedekind&#039;s Paradise</title>
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		<title>Dedekind&#039;s Paradise</title>
		<link>http://dedekindsparadise.wordpress.com</link>
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		<title>Part 1 &#8211; The main ideas of Global Class Field Theory</title>
		<link>http://dedekindsparadise.wordpress.com/2012/03/05/part-1-the-main-ideas-of-global-class-field-theory/</link>
		<comments>http://dedekindsparadise.wordpress.com/2012/03/05/part-1-the-main-ideas-of-global-class-field-theory/#comments</comments>
		<pubDate>Mon, 05 Mar 2012 11:03:23 +0000</pubDate>
		<dc:creator>danielfretwell</dc:creator>
				<category><![CDATA[Global Class Field Theory]]></category>

		<guid isPermaLink="false">http://dedekindsparadise.wordpress.com/?p=280</guid>
		<description><![CDATA[I really wanted to get started on something meaty so apologies to people following my posts on the basics of algebraic number theory&#8230;I will try and sort it out sometime in the future. This is actually going to be a short post compared to the usual. I wanted to start my exposition of class field [&#8230;]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=dedekindsparadise.wordpress.com&#038;blog=22263485&#038;post=280&#038;subd=dedekindsparadise&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
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		<title>Basic Notions part 2 &#8211; Ideals and the like!</title>
		<link>http://dedekindsparadise.wordpress.com/2012/02/29/basic-notions-part-2-ideals-and-the-like/</link>
		<comments>http://dedekindsparadise.wordpress.com/2012/02/29/basic-notions-part-2-ideals-and-the-like/#comments</comments>
		<pubDate>Wed, 29 Feb 2012 10:31:46 +0000</pubDate>
		<dc:creator>danielfretwell</dc:creator>
				<category><![CDATA[Uncategorized]]></category>

		<guid isPermaLink="false">http://dedekindsparadise.wordpress.com/?p=252</guid>
		<description><![CDATA[Every once in a while mathematicians have a breakthrough, and this completely revolutionizes maths&#8230;opening up a completely new way to view things. It is my aim in this post to explain one of these moments in history! We are all familliar with how factorisation works in . Take an integer, we may factorise it into [&#8230;]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=dedekindsparadise.wordpress.com&#038;blog=22263485&#038;post=252&#038;subd=dedekindsparadise&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
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		<title>Wilson&#8217;s theorem and further thought&#8230;</title>
		<link>http://dedekindsparadise.wordpress.com/2012/01/30/wilsons-theorem-and-further-thought/</link>
		<comments>http://dedekindsparadise.wordpress.com/2012/01/30/wilsons-theorem-and-further-thought/#comments</comments>
		<pubDate>Mon, 30 Jan 2012 20:23:11 +0000</pubDate>
		<dc:creator>danielfretwell</dc:creator>
				<category><![CDATA[Elementary Number Theory]]></category>

		<guid isPermaLink="false">http://dedekindsparadise.wordpress.com/?p=238</guid>
		<description><![CDATA[In any course on Elementary Number Theory you meet Wilson&#8217;s Theorem. This says that for any prime we have that . How do we prove this? Well I have reshaped the usual proof in order to generalise. My claim: For any finite Abelian group, the product of all elements is equal to the product of 2-torsion [&#8230;]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=dedekindsparadise.wordpress.com&#038;blog=22263485&#038;post=238&#038;subd=dedekindsparadise&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
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		<title>Rationalizing the denominator</title>
		<link>http://dedekindsparadise.wordpress.com/2012/01/30/rationalizing-the-denominator/</link>
		<comments>http://dedekindsparadise.wordpress.com/2012/01/30/rationalizing-the-denominator/#comments</comments>
		<pubDate>Mon, 30 Jan 2012 19:58:11 +0000</pubDate>
		<dc:creator>danielfretwell</dc:creator>
				<category><![CDATA[Algebraic Number Theory]]></category>
		<category><![CDATA[Uncategorized]]></category>

		<guid isPermaLink="false">http://dedekindsparadise.wordpress.com/?p=231</guid>
		<description><![CDATA[Remember how last time I found that returning to old problems spurred on current thinking? Well it has happened again. This time returning to Alevel maths has turned up something interesting. We learn how to rationalize the denominator of a fraction involving square roots quite early on in the Alevel course. For most this is really just [&#8230;]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=dedekindsparadise.wordpress.com&#038;blog=22263485&#038;post=231&#038;subd=dedekindsparadise&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>0</slash:comments>
	
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			<media:title type="html">danielfretwell</media:title>
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		<title>Bijective Integers.</title>
		<link>http://dedekindsparadise.wordpress.com/2012/01/09/bijective-integers/</link>
		<comments>http://dedekindsparadise.wordpress.com/2012/01/09/bijective-integers/#comments</comments>
		<pubDate>Mon, 09 Jan 2012 11:43:39 +0000</pubDate>
		<dc:creator>danielfretwell</dc:creator>
				<category><![CDATA[Elementary Number Theory]]></category>
		<category><![CDATA[Number Theory]]></category>

		<guid isPermaLink="false">http://dedekindsparadise.wordpress.com/?p=203</guid>
		<description><![CDATA[Sometimes we return to old problems in order to solve them in quicker or nicer ways. I have just done this. When I was an undergrad I was playing around with powers in residue systems. Take any and consider the numbers  in . I was interested in the values of  that would make this set the biggest [&#8230;]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=dedekindsparadise.wordpress.com&#038;blog=22263485&#038;post=203&#038;subd=dedekindsparadise&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
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			<media:title type="html">danielfretwell</media:title>
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		<title>Limitations of RSA.</title>
		<link>http://dedekindsparadise.wordpress.com/2011/07/24/limitations-of-rsa/</link>
		<comments>http://dedekindsparadise.wordpress.com/2011/07/24/limitations-of-rsa/#comments</comments>
		<pubDate>Sun, 24 Jul 2011 15:48:53 +0000</pubDate>
		<dc:creator>danielfretwell</dc:creator>
				<category><![CDATA[Cryptography]]></category>

		<guid isPermaLink="false">http://dedekindsparadise.wordpress.com/?p=185</guid>
		<description><![CDATA[RSA, as we saw is a really amazing public key cipher that uses only basic number theory in its description. However, whenever a new cipher appears there will be many people that test its security and whenever possible will try to break it. So far RSA has not been broken but certain bad things can happen [&#8230;]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=dedekindsparadise.wordpress.com&#038;blog=22263485&#038;post=185&#038;subd=dedekindsparadise&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>0</slash:comments>
	
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			<media:title type="html">danielfretwell</media:title>
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		<title>What is RSA and how does it work?</title>
		<link>http://dedekindsparadise.wordpress.com/2011/07/24/what-is-rsa-and-how-does-it-work/</link>
		<comments>http://dedekindsparadise.wordpress.com/2011/07/24/what-is-rsa-and-how-does-it-work/#comments</comments>
		<pubDate>Sun, 24 Jul 2011 14:48:18 +0000</pubDate>
		<dc:creator>danielfretwell</dc:creator>
				<category><![CDATA[Cryptography]]></category>

		<guid isPermaLink="false">http://dedekindsparadise.wordpress.com/?p=165</guid>
		<description><![CDATA[People always say to me, &#8220;Dan, what is number theory even needed for?&#8221; Usually I give one of two responses, either &#8220;Does it have to have a use?&#8221; or &#8220;One major use is in cryptography&#8221;. I approach the second of these responses here. Number theory is one of the oldest branches of mathematics that arose [&#8230;]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=dedekindsparadise.wordpress.com&#038;blog=22263485&#038;post=165&#038;subd=dedekindsparadise&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>0</slash:comments>
	
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			<media:title type="html">danielfretwell</media:title>
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		<title>Why is Quadratic Reciprocity so b-e-a-utiful?</title>
		<link>http://dedekindsparadise.wordpress.com/2011/07/02/why-is-quadratic-reciprocity-so-b-e-a-utiful/</link>
		<comments>http://dedekindsparadise.wordpress.com/2011/07/02/why-is-quadratic-reciprocity-so-b-e-a-utiful/#comments</comments>
		<pubDate>Sat, 02 Jul 2011 10:37:24 +0000</pubDate>
		<dc:creator>danielfretwell</dc:creator>
				<category><![CDATA[Elementary Number Theory]]></category>

		<guid isPermaLink="false">http://dedekindsparadise.wordpress.com/?p=120</guid>
		<description><![CDATA[This year I was able to do some tutorial teaching at uni. I was assigned a class of students doing the (extremely popular) course on Elementary Number Theory. About halfway through the course the students learn about quadratic residues and quadratic reciprocity. However, when the students turned up to tutorials the beauty of these topics were not transfered. [&#8230;]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=dedekindsparadise.wordpress.com&#038;blog=22263485&#038;post=120&#038;subd=dedekindsparadise&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
		<wfw:commentRss>http://dedekindsparadise.wordpress.com/2011/07/02/why-is-quadratic-reciprocity-so-b-e-a-utiful/feed/</wfw:commentRss>
		<slash:comments>2</slash:comments>
	
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			<media:title type="html">danielfretwell</media:title>
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		<title>Basic notions pt 1 &#8211; Number fields and rings of integers</title>
		<link>http://dedekindsparadise.wordpress.com/2011/04/19/basic-notions-pt-1-number-fields-and-rings-of-integers/</link>
		<comments>http://dedekindsparadise.wordpress.com/2011/04/19/basic-notions-pt-1-number-fields-and-rings-of-integers/#comments</comments>
		<pubDate>Tue, 19 Apr 2011 12:32:02 +0000</pubDate>
		<dc:creator>danielfretwell</dc:creator>
				<category><![CDATA[Algebraic Number Theory]]></category>

		<guid isPermaLink="false">http://dedekindsparadise.wordpress.com/?p=92</guid>
		<description><![CDATA[So yeah,  now that we have seen a little of what algebraic number theory is about, let&#8217;s get started properly. Usually in number theory we  love to work with integers, or at a push rational numbers. This is not really enough to solve some problems. A fact that you all know is that really each rational number can [&#8230;]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=dedekindsparadise.wordpress.com&#038;blog=22263485&#038;post=92&#038;subd=dedekindsparadise&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
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			<media:title type="html">danielfretwell</media:title>
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		<title>What is algebraic number theory?</title>
		<link>http://dedekindsparadise.wordpress.com/2011/04/18/what-is-algebraic-number-theory/</link>
		<comments>http://dedekindsparadise.wordpress.com/2011/04/18/what-is-algebraic-number-theory/#comments</comments>
		<pubDate>Mon, 18 Apr 2011 10:02:34 +0000</pubDate>
		<dc:creator>danielfretwell</dc:creator>
				<category><![CDATA[Algebraic Number Theory]]></category>

		<guid isPermaLink="false">http://dedekindsparadise.wordpress.com/?p=25</guid>
		<description><![CDATA[This is quite a difficult question to answer precisely. This branch of maths has evolved so much over the last or so years. On a basic level it is two things. On the one hand it is essentially the study of number theory by using algebraic methods (groups, rings, fields, modules etc). On the other [&#8230;]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=dedekindsparadise.wordpress.com&#038;blog=22263485&#038;post=25&#038;subd=dedekindsparadise&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
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