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This is not hard to see: if K was finite, K = {0, 1, α3 , . . , αn }, we could write down a polynomial, say, f (x) = x(x − 1)(x − α1 ) · · · (x − αn ) + 1 which has no roots in K. 34. Suppose C is defined by f (x, y) = a0 (x) + · · · + an (x)y n ∈ K[x, y], for n ≥ 1. For any α ∈ K the polynomial f (α, y) lies in K[y] and hence must have n roots, counting multiplicities. Since K is infinite we obtain infinitely many points (α, β) on C, where α is arbitrary and β is root of f (α, y). The case n = 0 is left for you.

Geometrically this means that L3 is tangent to C at (1 : 0 : 1). e. their intersection with Uz where the points have real coordinates. This can be easily obtained by dehomogenizing the equations of the curve and the lines. 5. Intersection of a conic and three lines. 1. Tangent Lines. By definition a line L is tangent to a curve C at a point p0 if L intersects C at p0 with multiplicity greater than one. We will write this in coordinates and derive an equation of the tangent line to C at p0 . Let C be a projective curve in P2 given by a homogeneous polynomial F (x, y, z).

Let F = R, the real numbers. (a) f (x, y) = x2 − y 2 = (x − y)(x + y). The curve C is the union of two lines y = x and y = −x, which are the absolutely irreducible components of C. (b) f (x, y) = x2 + y 2 − 1. The curve C is the unit circle. 31. (c) f (x, y) = a0 (x). The irreducible components of C are vertical lines x = α, for every real root α of a0 (x). The absolutely irreducible components are the vertical lines x = α for every complex root α of a0 (x). We would like to have a one-to-one correspondence between irreducible curves C and their defining polynomials f ∈ F[x, y], up to a constant multiple.

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Algebraic Curves and Codes [Lecture notes] by Ivan Soprunov

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