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	<title>Tom Lovering&#039;s Blog</title>
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	<link>http://tlovering.wordpress.com</link>
	<description>For Mathematical Musings</description>
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		<title>Tom Lovering&#039;s Blog</title>
		<link>http://tlovering.wordpress.com</link>
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		<item>
		<title>Why Modularity implies Fermat&#8217;s Last Theorem</title>
		<link>http://tlovering.wordpress.com/2012/01/08/why-modularity-implies-fermats-last-theorem/</link>
		<comments>http://tlovering.wordpress.com/2012/01/08/why-modularity-implies-fermats-last-theorem/#comments</comments>
		<pubDate>Sun, 08 Jan 2012 16:30:57 +0000</pubDate>
		<dc:creator>tlovering</dc:creator>
				<category><![CDATA[Mathematics]]></category>

		<guid isPermaLink="false">http://tlovering.wordpress.com/?p=610</guid>
		<description><![CDATA[In this post we shall sketch the reduction of Fermat&#8217;s Last Theorem to Wiles&#8217; Theorem that every semistable elliptic curve over is modular. I believe the ideas are due mainly to Frey, with a helpful big black box courtesy of Serre and Ribet. My main reference is the article of Stephens from the 1995 conference. [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=tlovering.wordpress.com&amp;blog=5997865&amp;post=610&amp;subd=tlovering&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
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		<media:content url="http://1.gravatar.com/avatar/1fd33d1d6e57765bb34444a6a2d13b9f?s=96&#38;d=http%3A%2F%2F1.gravatar.com%2Favatar%2Fad516503a11cd5ca435acc9bb6523536%3Fs%3D96" medium="image">
			<media:title type="html">Tom Lovering</media:title>
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		<item>
		<title>Hilbert 90 from 1-Dimensional Galois Descent</title>
		<link>http://tlovering.wordpress.com/2011/12/31/hilbert-90-from-1-dimensional-galois-descent/</link>
		<comments>http://tlovering.wordpress.com/2011/12/31/hilbert-90-from-1-dimensional-galois-descent/#comments</comments>
		<pubDate>Sat, 31 Dec 2011 17:40:34 +0000</pubDate>
		<dc:creator>tlovering</dc:creator>
				<category><![CDATA[Mathematics]]></category>

		<guid isPermaLink="false">http://tlovering.wordpress.com/2011/12/31/hilbert-90-from-1-dimensional-galois-descent/</guid>
		<description><![CDATA[In this short post I wish to record the proof of Hilbert&#8217;s Theorem 90 by interpreting it as the 1-dimensional case of Galois descent. The ideas all come from the `Arcata&#8217; in SGA 4 1/2. Note that in some sense this completes the Kummer Theory argument in the previous post, except now I am assuming [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=tlovering.wordpress.com&amp;blog=5997865&amp;post=606&amp;subd=tlovering&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
		<wfw:commentRss>http://tlovering.wordpress.com/2011/12/31/hilbert-90-from-1-dimensional-galois-descent/feed/</wfw:commentRss>
		<slash:comments>1</slash:comments>
	
		<media:content url="http://1.gravatar.com/avatar/1fd33d1d6e57765bb34444a6a2d13b9f?s=96&#38;d=http%3A%2F%2F1.gravatar.com%2Favatar%2Fad516503a11cd5ca435acc9bb6523536%3Fs%3D96" medium="image">
			<media:title type="html">Tom Lovering</media:title>
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		<item>
		<title>Basic Applications of Elementary Galois Cohomology</title>
		<link>http://tlovering.wordpress.com/2011/12/30/basic-applications-of-elementary-galois-cohomology/</link>
		<comments>http://tlovering.wordpress.com/2011/12/30/basic-applications-of-elementary-galois-cohomology/#comments</comments>
		<pubDate>Fri, 30 Dec 2011 20:40:12 +0000</pubDate>
		<dc:creator>tlovering</dc:creator>
				<category><![CDATA[Mathematics]]></category>

		<guid isPermaLink="false">http://tlovering.wordpress.com/2011/12/30/basic-applications-of-elementary-galois-cohomology/</guid>
		<description><![CDATA[In this post we shall develop the very basics of Galois cohomology, see two basic but important applications: Kummer Theory and the Weak Mordell-Weil Theorem, and remark on the general shape of both arguments. Hopefully at a later point I will post on Neukirch&#8217;s `General&#8217; Class Field Theory which is perhaps a more complex realisation [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=tlovering.wordpress.com&amp;blog=5997865&amp;post=595&amp;subd=tlovering&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>0</slash:comments>
	
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			<media:title type="html">Tom Lovering</media:title>
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		<item>
		<title>The Etale Cohomology of Elliptic Curves</title>
		<link>http://tlovering.wordpress.com/2011/12/04/the-etale-cohomology-of-elliptic-curves/</link>
		<comments>http://tlovering.wordpress.com/2011/12/04/the-etale-cohomology-of-elliptic-curves/#comments</comments>
		<pubDate>Sun, 04 Dec 2011 04:04:50 +0000</pubDate>
		<dc:creator>tlovering</dc:creator>
				<category><![CDATA[Mathematics]]></category>

		<guid isPermaLink="false">http://tlovering.wordpress.com/?p=579</guid>
		<description><![CDATA[As a few people have requested them, here are the notes from my recent seminar on the Etale Cohomology of elliptic curves. They definitely contain a cheat or two, and aren&#8217;t particularly well formatted (and indeed aren&#8217;t really the same as what I actually said in the talk), but here they are anyway. I should [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=tlovering.wordpress.com&amp;blog=5997865&amp;post=579&amp;subd=tlovering&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>0</slash:comments>
	
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			<media:title type="html">Tom Lovering</media:title>
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		<title>Surjective Maps of Schemes (over non-algebraically closed fields)</title>
		<link>http://tlovering.wordpress.com/2011/11/27/surjective-maps-of-schemes-over-non-algebraically-closed-fields/</link>
		<comments>http://tlovering.wordpress.com/2011/11/27/surjective-maps-of-schemes-over-non-algebraically-closed-fields/#comments</comments>
		<pubDate>Sun, 27 Nov 2011 15:28:22 +0000</pubDate>
		<dc:creator>tlovering</dc:creator>
				<category><![CDATA[Mathematics]]></category>

		<guid isPermaLink="false">http://tlovering.wordpress.com/?p=573</guid>
		<description><![CDATA[The notion of a surjective map of schemes is somewhat confusing. For example, consider an elliptic curve over of positive rank. The multiplication-by- map is obviously not surjective as a map , but in fact it is surjective as a map of schemes, because schemes carry baggage associated to the points we haven&#8217;t yet `added [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=tlovering.wordpress.com&amp;blog=5997865&amp;post=573&amp;subd=tlovering&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
		<wfw:commentRss>http://tlovering.wordpress.com/2011/11/27/surjective-maps-of-schemes-over-non-algebraically-closed-fields/feed/</wfw:commentRss>
		<slash:comments>3</slash:comments>
	
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			<media:title type="html">Tom Lovering</media:title>
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		<item>
		<title>Global Class Field Theory 3</title>
		<link>http://tlovering.wordpress.com/2011/11/09/global-class-field-theory-3/</link>
		<comments>http://tlovering.wordpress.com/2011/11/09/global-class-field-theory-3/#comments</comments>
		<pubDate>Wed, 09 Nov 2011 22:18:36 +0000</pubDate>
		<dc:creator>tlovering</dc:creator>
				<category><![CDATA[Mathematics]]></category>

		<guid isPermaLink="false">http://tlovering.wordpress.com/?p=564</guid>
		<description><![CDATA[In the third post of this series, we shall very briefly introduce the formalism of ideles and restate the Artin reciprocity theorem in our new language, following the ideas of Chevalley. It is a little difficult to get used to the language of adeles and ideles, but they turn out to be a very natural [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=tlovering.wordpress.com&amp;blog=5997865&amp;post=564&amp;subd=tlovering&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>0</slash:comments>
	
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			<media:title type="html">Tom Lovering</media:title>
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		<item>
		<title>Zeta functions of varieties over a finite field</title>
		<link>http://tlovering.wordpress.com/2011/11/01/zeta-functions-of-varieties-over-a-finite-field/</link>
		<comments>http://tlovering.wordpress.com/2011/11/01/zeta-functions-of-varieties-over-a-finite-field/#comments</comments>
		<pubDate>Tue, 01 Nov 2011 16:04:37 +0000</pubDate>
		<dc:creator>tlovering</dc:creator>
				<category><![CDATA[Mathematics]]></category>

		<guid isPermaLink="false">http://tlovering.wordpress.com/?p=559</guid>
		<description><![CDATA[In this short post we look at the general definition of a zeta function for a scheme of finite type over and record its relationship with the zeta function of an algebraic variety over a finite field , defined as a kind of generating function for the number of points of over all finite extensions [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=tlovering.wordpress.com&amp;blog=5997865&amp;post=559&amp;subd=tlovering&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
		<wfw:commentRss>http://tlovering.wordpress.com/2011/11/01/zeta-functions-of-varieties-over-a-finite-field/feed/</wfw:commentRss>
		<slash:comments>0</slash:comments>
	
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			<media:title type="html">Tom Lovering</media:title>
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		<item>
		<title>Global Class Field Theory 2</title>
		<link>http://tlovering.wordpress.com/2011/10/21/global-class-field-theory-2/</link>
		<comments>http://tlovering.wordpress.com/2011/10/21/global-class-field-theory-2/#comments</comments>
		<pubDate>Fri, 21 Oct 2011 21:20:36 +0000</pubDate>
		<dc:creator>tlovering</dc:creator>
				<category><![CDATA[Mathematics]]></category>

		<guid isPermaLink="false">http://tlovering.wordpress.com/?p=544</guid>
		<description><![CDATA[Last time, we were thinking about an abelian extension , and we constructed, for most of the primes of a well-defined associated Frobenius element . To do this, we used the canonical isomorphism between the decomposition group of some prime extending (a subgroup of ) and the Galois group of the residue fields. Since this [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=tlovering.wordpress.com&amp;blog=5997865&amp;post=544&amp;subd=tlovering&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>0</slash:comments>
	
		<media:content url="http://1.gravatar.com/avatar/1fd33d1d6e57765bb34444a6a2d13b9f?s=96&#38;d=http%3A%2F%2F1.gravatar.com%2Favatar%2Fad516503a11cd5ca435acc9bb6523536%3Fs%3D96" medium="image">
			<media:title type="html">Tom Lovering</media:title>
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		<item>
		<title>Analytic proof that cyclotomic polynomials are irreducible</title>
		<link>http://tlovering.wordpress.com/2011/10/17/analytic-proof-that-cyclotomic-polynomials-are-irreducible/</link>
		<comments>http://tlovering.wordpress.com/2011/10/17/analytic-proof-that-cyclotomic-polynomials-are-irreducible/#comments</comments>
		<pubDate>Mon, 17 Oct 2011 21:32:11 +0000</pubDate>
		<dc:creator>tlovering</dc:creator>
				<category><![CDATA[Mathematics]]></category>

		<guid isPermaLink="false">http://tlovering.wordpress.com/?p=548</guid>
		<description><![CDATA[Apparently this is due to Kronecker &#8211; the idea is to prove that cyclotomic polynomials are irreducible indirectly using the following two analytic facts: Fact 1 (Dirichlet&#8217;s theorem, special case): The Dirichlet density of primes which are congruent to modulo is precisely . Fact 2 (due to Kronecker, but a super special case of Cebotarev&#8217;s [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=tlovering.wordpress.com&amp;blog=5997865&amp;post=548&amp;subd=tlovering&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
		<wfw:commentRss>http://tlovering.wordpress.com/2011/10/17/analytic-proof-that-cyclotomic-polynomials-are-irreducible/feed/</wfw:commentRss>
		<slash:comments>0</slash:comments>
	
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			<media:title type="html">Tom Lovering</media:title>
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		<item>
		<title>Global Class Field Theory 1</title>
		<link>http://tlovering.wordpress.com/2011/10/16/global-class-field-theory-1/</link>
		<comments>http://tlovering.wordpress.com/2011/10/16/global-class-field-theory-1/#comments</comments>
		<pubDate>Sun, 16 Oct 2011 12:58:37 +0000</pubDate>
		<dc:creator>tlovering</dc:creator>
				<category><![CDATA[Mathematics]]></category>

		<guid isPermaLink="false">http://tlovering.wordpress.com/?p=542</guid>
		<description><![CDATA[In this series of posts I shall try to sketch (at least statements of) the basic results of global class field theory, following John Tate&#8217;s notes from the 1966 Brighton Conference. I begin by emphasising that I am writing them mainly in order to try to learn the material myself rather than really as an [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=tlovering.wordpress.com&amp;blog=5997865&amp;post=542&amp;subd=tlovering&amp;ref=&amp;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>2</slash:comments>
	
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			<media:title type="html">Tom Lovering</media:title>
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