New PDF release: Algebraic Curves and Codes [Lecture notes]

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By Ivan Soprunov

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Additional resources for Algebraic Curves and Codes [Lecture notes]

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57. f, g ∈ K[x] have a common root in K if and only if there exist non-zero f1 , g1 ∈ K[x] such that f g1 = gf1 and deg f1 < deg f , deg g1 < deg g. Proof. (⇒) Suppose f, g have a common root α ∈ K. Then f = (x − α)f1 and g = (x − α)g1 for some f1 , g1 ∈ K with deg f1 < deg f , deg g1 < deg g. Also f g1 = gf1 from above. (⇐) Suppose h = f g1 = gf1 with deg f1 < deg f , deg g1 < deg g. Then the roots of f and the roots of g are also the roots of h. But deg h = deg f + deg g1 < deg f + deg g, hence the sets of the roots of f and g cannot be disjoint.

We will define a very important map from Fq to itself. 8. Let Fq be a finite field of characteristic p. The map σ : Fq → F q , is called the Frobenius automorphism. α �→ αp Here some of its properties. 9. Let Fq be a finite field of q = pn elements. Then (1) for any α, β ∈ Fq we have (α + β)p = αp + β p ; (2) the map α �→ αp is an automorphism of Fq which fixes Fp ; (3) the Galois group of all automorphisms of Fq which fix Fp , Gal(Fq ) = {φ : Fq → Fq | φ(a) = a, ∀a ∈ Fp }, is cyclic of order n, generated by σ.

Recall that in Uz we have u = xz , v = yz , so the two equations become yz − 2 xz = 0 and yz − 2 xz − 2 = 0. If we clear the denominators we obtain y − 2x = 0 and y − 2x − 2z = 0. Notice that now they make sense for z = 0 as well. Thus they define two lines in the projective space: ¯ 1 = {(x : y : z) ∈ P2 | y − 2x = 0}, L ¯ 2 = {(x : y : z) ∈ P2 | y − 2x − 2z = 0}, L which coincide with L1 and L2 on Uz . However each of them has one extra point ¯ 1 and L ¯ 2 intersect in P2 at one point (1 : 2 : 0).

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Algebraic Curves and Codes [Lecture notes] by Ivan Soprunov

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