The following problem is found in chapter 3 of Golan's linear algebra book.
Let $p$ be a prime. Let $V$ be a vector space over $F_p$, the field with $p$ elements. Show that $V$ is not the union of $k$ subspaces, for any $k\le p$.
Note that the field is not necessarily finite.
The problem is clearly incorrect as stated. If we require the subspaces to be distinct, nontrivial, and proper, does it become true? And if so, how might one go about solving it?