*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.

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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