Since is a Euclidean domain and thus in particular a PID. By Maximal ideal iff prime ideal in a PID, it follows is maximal and thus as defined is indeed a field ( for commutative is a field iff is maximal). Let denote the projection. Then

Since all we adjoined was , this is indeed simple.

Now suppose is an extension with , , so the Evaluation map vanishes at , whence and thus by the universal property of quotients there is a unique homomorphism

which gives the desired tower of extensions.