Let K = C(x, y), the field of rational functions in two variables over the complex numbers, and consider the algebraic curve defined by y² = x³ + x + 1. a) Prove that the curve defined by y² = x³ + x + 1 is nonsingular. Provide a detailed analysis of its singular points, if any. b) Determine the genus of the curve and explain the method used to compute it. c) Investigate whether K is a Galois extension of C(x). If it is, describe its Galois group explicitly. d) Explore the function field extension C(x, y)/C(x) by determining its degree and providing a basis for C(x, y) as a vector space over C(x). e) Using the Riemann-Roch theorem, compute the dimension of the space of global sections of a given divisor on the curve. Provide a specific example of such a divisor and perform the calculation.

Determine the number and nature of the roots of the equation in Exercise 67. Calculate the discriminant associated with the equation 6x2+5x+1=0.