Algebraic Curves and Codes [Lecture notes] by Ivan Soprunov

By Ivan Soprunov

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Can we parametrize the points of C just like we parametrize all the points of a line? In other words, can we write the coordinates (x, y) ∈ C as rational functions of a parameter t? The answer turns out to be yes, and Bezout’s theorem will help us here.

Note that in this case K is a vector space over F. An field extension F ⊂ K is called finite if K is a finite dimensional space over F. In particular, if K is a finite field then F ⊂ K is a finite field extension. In this case K must have a basis {v1 , . . e. K = {c1 v1 + · · · + ck vk | ci ∈ F}, k = dimF K. For example, F4 is a 2-dimensional vector space over F2 with a basis {1, α}. More generally, Fpn is an n-dimensional vector space over Fp with a basis {1, α, . . , αn−1 }. 2. Let F ⊂ K ⊂ L be a “tower” of field extensions and L is finite dimensional over F.

This may look redundant, but it will be handy later when we talk about the affine and the projective plane. 32. A plane affine curve C is the set C = {(x, y) ∈ A2 | f (x, y) = 0} for some non-constant polynomial f ∈ F[x, y]. The degree of C is the degree of the polynomial f . It is custom to call curves of degree two conics and curves of degree three cubics. Let us write f as a product of distinct (absolutely) irreducible factors f = f1k1 · · · fsks , where fi �= cfj for any i �= j, c ∈ F. Then C is a union of curves Ci = {(x, y) ∈ A2 | fi (x, y) = 0}, which are called the (absolutely) irreducible components of C.

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