Quadratic reciprocity has hundreds of proofs, but the nicest ones I’ve seen (at least at the elementary level) use Gauss sums. One variant uses the cyclotomic field ℚ(ζ), where ζ is a p-th root of unity. Another brings in the finite fields 𝔽p and 𝔽q.

I wrote up a long, loving, and chatty treatment several years ago, going through the details for several examples. Much longer than the proofs! The diagram up top may give you an inkling.

Anyway, here  it is.

Filed under Number Theory

Nonstandard Models of Arithmetic 31

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MW: Last time we learned about the “back-and-forth” condition for two countable structures M and N for a (countable) language L:

Filed under Conversations, Peano Arithmetic

Nonstandard Models of Arithmetic 30

MW: Time to finish off Enayat’s Theorem 7:

Theorem 7: Every countable recursively saturated model N of PA+ΦT is a T-standard model of PA.

Filed under Conversations, Peano Arithmetic

Nonstandard Models of Arithmetic 29

MW: We’re still going through Enayat’s proof of his Theorem 7:

Theorem 7: Every countable recursively saturated model N of PA+ΦT is a T-standard model of PA.

Filed under Conversations, Peano Arithmetic

Nonstandard Models of Arithmetic 28

MW: I ended the last post with a puzzle. Here it is again, in more detail.

Filed under Conversations, Peano Arithmetic

Nonstandard Models of Arithmetic 27

MW: Enayat’s second major result is:

Theorem 7: Every countable recursively saturated model of PA+ΦT is a T-standard model of PA.

Filed under Conversations, Peano Arithmetic

Very Unique

My unique (but not very unique) microwave

Everyone has their pet peeves, and peeves about language abound. My pet peeve is with people who object that “very unique” is illogical. For example, this pithy statement:

Uniqueness is a binary condition. Something is unique or it is not. There are no degrees of uniqueness. Something cannot be partly unique, mostly unique, very unique, etc.

Filed under Logic, Rants

Selections from the History of Astronomy

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Armillary Sphere, Facade of Santa Maria Novella (Wikimedia Commons)

Another post from the History Book Club, this time based on three books:

• To Explain the World: The Discovery of Modern Science, by Steven Weinberg.
• The Sun in the Church: Cathedrals as Solar Observatories, by J.L. Heilbron.
• The Composition of Kepler’s Astronomia Nova, by James Voelkel.

Filed under Astronomy, History Book Club