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Fearless Symmetry by Robert Gross, Avner Ash

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In this chapter we perform a lot of algebra to reinterpret
quadratic reciprocity. As explained in chapter 7, quadratic
reciprocity appears to be an innocuous curiosity, the
main part of which states that if p and q are two odd
primes, then the “squareness” of p modulo q and the
“squareness” of q modulo p are related in a certain pre-
dictable way. Throughout the nineteenth century, number
theorists proved similar theorems about “cubedness” and
higher powers. In the beginning of the twentieth century,
mathematicians began to realize that all of these power
reciprocity theorems could be clarified and unified in
terms of Galois representations. The key was to think of a
reciprocity law as a way of describing χ
φ
(Frob
p
) by other
means, as we explained in chapter 17.
Here we undertake this interpretation for quadratic
reciprocity—the simplest, the oldest, and the most fun-
damental of all reciprocity laws. We also must bring
Z-equations into the discussion, particularly the equation
x
2
W = 0 for various integers W. This is because asking
about the “squareness” of p modulo q is the same as asking
whether x
2
p = 0 has a solution modulo q.
The black boxes we use in this chapter are introduced
in the first few paragraphs. As we go through the algebra,
we will be able to see into these black boxes, another sign
that quadratic reciprocity is relatively easy.

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