All Questions

Let $P_1,\dots,P_k$ be polynomials over $\mathbf{C}$, no two of them being proportional. Does there exist an integer $N$ such that $P_1^N,\dots,P_k^N$ are linearly independent?

In functional analysis, there are many examples of things that "go wrong" in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a ...

Is there a field which is the union of finitely many proper subfields?

The title of my Q. says it all: QUESTION:   Who defined and who coined: module? Would it be Emmy Noether? EDIT   In view of @anon's and KConrad's answers, and as it could have been ...