A parameter space for Tschirnhaus transformations

In this thesis we explore the topology of a space $\T_n$ parameterizing Tschirnhaus transformations. We begin with an exposition of Tschirnhaus transformations in modern language, and a demonstration of their use in solving the depressed cubic equation. We then construct $\T_n$ and show that it has the structure of a $\PConf_n(\C)$ bundle over $\UConf_n(\C)$. We see that rational maps from a variety $Y$ into $\T_n$ correspond to primitive elements in degree $n$ field extensions of the function field $K(Y)$. We give a formula for the rational cohomology of $\T_n$, see that it is homologically stable as $n \to \infty$, and explicitly compute the dimensions of the stable cohomology groups in low degree, along the way computing character polynomials for $H^i(P_n; \Q)$ as an $S_n$ representation for $i \le 5$. Finally, we give a brief historical overview of progress in solving low degree polynomials, leading into the active research fields of essential dimension, resolvent degree and the algebraic form of Hilbert's 13th problem, topics to which we hope this parameter space and our results on it might apply and prove useful.