A Note on Diem's Proof

Semaev’s summation polynomials are suggested to be used to construct Index Calculus for elliptic curves over extension fields. The complexity of the proposed algorithm was first studied by Diem with the help of Weil restriction and intersection theory. The key ingredient is to analyse the probability that a uniformly distributed point has an isolated decomposition over the factor base. Following his tactics, we present his result in a simple manner by recourse to the theory of function fields and generic resultant.