Heron’s formula

Heron’s formula for the area of a triangle with side lengths is where is the semiperimeter. Today I’d like to try to prove this using as little geometry as possible.

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Ideals and the category of commutative rings

In this post I’d like to give a better (by which I mean category-theoretic) definition of the lattice of ideals than the standard one. We know that the lattice of ideals has meets and joins defined by intersection and sum, respectively, and that if a lattice is viewed as a category whose arrows are the [...]

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Some quadratic reciprocity

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. [...]

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The arithmetic plane

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 [...]

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