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Two elements r1 and r2 of R are called associates if there is a unit u ∈ R such that r1 = ur2. 2. The prime elements of an integral domain are irreducible. Proof. Let p be a prime element of an integral domain R. Then, for any a, b ∈ R such that p = ab we have that p|a or p|b. Assume, without loss of generality, that p|a. Then there is c ∈ R such that a = pc. Hence, is a unit and therefore p is irreducible. Definition An integral domain R in which every non-zero element a can be written uniquely with respect to a unit as a product of irreducible elements of R is called a unique factorization domain.

3 Riemann-Roch Space Definition Let F/K(x) be a function field and D a divisor of F. The set is called Riemann-Roch space of D. The following lemma justifies why L (D) is called a space. 1. Let F/K(x)be a function field. For any divisor D the Riemann-Roch space L(D) is a K-vector space. Proof. Let a, b ∈ L(D) and k ∈ K. For every place of F we have that and Thus, a + b and ka belong to L(D) and therefore L(D) is a K -vector space. Suppose that where Pi are the zeros, Qj are the poles and ni, mj ∈ .

The second partial derivative for x = 1 and y = 2 is not equal to 0. Therefore, the multiplicity of P = (1 : 2 : 0) is 2. In order to find if the singularity is ordinary we need to compute the sum Computation gives us the homogeneous polynomial It is easy to see that the above polynomial is not a perfect square and therefore the curve has an ordinary singularity at P(1 : 2 : 0). Since the unique singular point of the curve is ordinary, we compute the genus of the curve for d = 3 and m = 2 by definition and we get that So, actually the curve is a rational curve.