Abstract: In the mid-20th century, Weil developed, and Grothendieck generalized, a technique called descent. Using descent, one is able to answer the following question: given a non algebraically closed field k, and a variety X defined over the algebraic closure of k, can we find a variety Y/k whose base extension to the algebraic closure is isomorphic to X?

After reviewing the basics of group cohomology and twisting, I will show how the descent principle allows us to easily conclude that a certain cohomology group vanishes, which in turn allows us to easily prove a famous theorem in Galois theory, which we can then use to parametrize all solutions to a certain class of diophantine equations.