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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Primes and ideals

Probably the first important result in algebraic number theory is the following. Let be a finite field extension of . Let be the ring of algebraic integers in . Theorem: The ideals of factor uniquely into prime ideals. This is the “correct” generalization of the fact that , as well as some small extensions of [...]

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