Differential Essential Dimension
differential Hopf-Galois extension
Roughly speaking, the essential dimension of an algebraic object is the minimum number of parameters needed to specify the object. It was first introduced by J. Buhler and Z. Reichstein where it was used to bound the number of parameters that one may eliminate from a general polynomial by means of Tschirnhaus transformations. In this thesis we define an analogue of essential dimension in differential algebra. As application, we show that the number of parameters in a general homogeneous linear differential equation over a field cannot be reduced via gauge transformations over the given field. We also bound the number of parameters needed to describe certain generic Picard-Vessiot extensions.