*Road Map*

Adding elements to our number systems is not sufficient to make the two definitions of degree agree. Next, we must add elements to the plane, changing the *affine plane* to the *projective plane*, to force there to be more intersection points between a line and a curve.

The story so far: If we have a polynomial *f*(*x*, *y*) in two variables of degree *d*, and if we intersect the graph of *f* = 0 with a parametrized line given by the simultaneous equations *x* = *at* + *b*, *y* = *ct* + *e*, we get a number *N* of intersection points. This number *N* is equal to the number of solutions of the polynomial equation ...

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