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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.51 Local and Global in Number Theory

   Fernando Q. Gοuνêa

Analogy is a powerful tool. When one can see parallels between two different theories, this often allows one to transport insights from one to the other. The idea of studying something “locally” comes from the theory of functions. Imported into number theory by way of an analogy between functions and numbers, it leads us to a whole new kind of number, the p-adic numbers, and to the local-global principle, which has become one of the guiding ideas of modern number theory.

1 Studying Functions Locally

Suppose that we have a polynomial such as

f(x) = -18 + 21x - 26x2 + 22x3 - 8x4 + x5.

From the very way the polynomial is written down, we can see certain things about it. For example, ...

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