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Elliptic Tales by Robert Gross, Avner Ash

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Chapter 2

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ALGEBRAIC CLOSURES

Road Map

We saw above that our two definitions of degree might not agree. We would like to force them to agree, and the first step in that task is to add elements to our number systems. Adding these “imaginary” numbers will mean that a polynomial equation in a single variable will always have at least one solution.

1. Square Roots of Minus One

The story so far: If we have a polynomial f(x, y) in two variables, we have defined the degree d of f to be the maximum of the degrees of its monomials. If we intersect the graph of f = 0 with a parametrized line in the parameter t, that is, a line given by the simultaneous ...

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