In the previous post we showed that the splitting behavior of a rational prime in the ring of cyclotomic integers depends only on the residue class of . This is suggestive enough of quadratic reciprocity that now would be a good time to give a full proof. The key result is the following fundamental observation. [...]
Posts Tagged ‘Galois theory’
Some quadratic reciprocity
Posted in algebraic number theory, tagged Fourier transforms, Frobenius map, Galois theory on January 11, 2010 | 2 Comments »
The arithmetic plane
Posted in algebraic number theory, arithmetic geometry, tagged finite fields, Frobenius map, Galois theory on January 4, 2010 | 1 Comment »
If you haven’t seen them already, you might want to read John Baez’s week205 and Lieven le Bruyn’s series of posts on the subject of spectra. I especially recommend that you take a look at the picture of to which Lieven le Bruyn links before reading this post. John Baez’s introduction to week205 would probably [...]
The ideal-variety correspondence
Posted in algebraic geometry, commutative algebra, tagged adjoint functors, Galois theory, MaBloWriMo, Nullstellensatz on November 30, 2009 | Leave a Comment »
I guess I didn’t plan this very well! Instead of completing one series I ended one and am right in the middle of another. Well, I’d really like to continue this series, but seeing as how finals are coming up I probably won’t be able to maintain the one-a-day pace. So I’ll just stop tagging [...]
Some adjoint functors
Posted in category theory, representation theory, tagged abstract nonsense, adjoint functors, Galois theory, universal properties on October 27, 2009 | 4 Comments »
Note: as usual, I will be playing free and loose with category theory in this post. Apologies to those who know better. One way to define a subgroup of a group is as the image of a homomorphism into . Given the inclusion map , the functor in the category of groups acts contravariantly to [...]
IMO 2009 and proof systems
Posted in abstract algebra, number theory, Putnam / competitions, remarks, tagged Chebyshev polynomials, equivalence relations, Galois theory, Grobner bases, pedagogy, philosophy of mathematics, trigonometry on July 17, 2009 | 3 Comments »
The problems from IMO 2009 are now available. I haven’t had much time to work on them, though. There are two classical geometry problems, which I already know I won’t attempt. While I am well aware that classical geometry often requires a great deal of ingenuity, I am also aware of the existence of the [...]
The magic of the Frobenius map II
Posted in abstract algebra, combinatorics, graph theory, number theory, tagged finite fields, Frobenius map, Galois theory, walks on graphs on June 9, 2009 | 3 Comments »
Once upon a time I discussed some interesting uses of the Frobenius map to solve some Putnam-style problems. Unfortunately, I wrote that post before becoming really interested in combinatorics, so I neglected to develop that particular side of the story, which I’d like to do now. The beginning of this story is the folklore combinatorial [...]
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