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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