A surjection problem leading to the Ax-Grothendieck theorem

摘要

We give a complete functional calculus for matrices of order two. The key tools are two simple Lemmas (Lemma 2.1 and Lemma 2.2, below), which are elementary but are somewhat new. As an application, we find the set of all polynomials P for which the matrix map X - P(X) on M(2, C) is surjective. The C-4-version of the famous Ax-Grothendieck theorem is obtained as an immediate Corollary.