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Z[sqrt{-3}] is the Eisenstein integers glued together at two points
Today’s post is a record of a very small observation from my time at PROMYS this summer. Below, by I mean a commutative ring regarded as an object in the opposite category . Once you’ve learned how to...
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Small factors in random polynomials over a finite field
Previously I mentioned very briefly Granville’s The Anatomy of Integers and Permutations, which explores an analogy between prime factorizations of integers and cycle decompositions of permutations....
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The p-group fixed point theorem
The goal of this post is to collect a list of applications of the following theorem, which is perhaps the simplest example of a fixed point theorem. Theorem: Let be a finite -group acting on a finite...
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Topological Diophantine equations
The problem of finding solutions to Diophantine equations can be recast in the following abstract form. Let be a commutative ring, which in the most classical case might be a number field like or the...
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The puzzle of Galois descent
Suppose we have a system of polynomial equations over a perfect (to keep things simple) field , and we’d like to consider solutions of it over various field extensions of . Write for the set of all...
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Fixed points of group actions on categories
Previously we described what it means for a group to act on a category (although we needed to slightly correct our initial definition). Today, as the next step in our attempt to understand Galois...
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Stating Galois descent
After a relaxing and enjoyable break, we’re finally in a position to state what it means for structures to satisfy Galois descent. Fix a field . The gadgets we want to study assign to each separable...
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Forms and Galois cohomology
Yesterday we gave a brief and abstract description of Galois descent, the punchline of which was that Galois descent could abstractly be described as a natural equivalence where is a Galois extension,...
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Separable algebras
Let be a commutative ring and let be a -algebra. In this post we’ll investigate a condition on which generalizes the condition that is a finite separable field extension (in the case that is a field)....
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The man who knew elliptic integrals, prime number theorems, and black holes
I went to see The Man Who Knew Infinity yesterday. I have nothing much to say about the movie as a movie that wasn’t already said in Scott Aaronson‘s review, except that I learned a few fun facts...
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