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If a field $F$ is such that $\left|F\right|>n-1$ why is $V$ a vector space over $F$ not equal to the union of proper subspaces of $V$

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?

  • $\begingroup$ I'm sure I have answered this somewhere in this forum :-) $\endgroup$ – Jyrki Lahtonen May 25 '12 at 7:58
  • $\begingroup$ @Jyrki: You did :-) $\endgroup$ – joriki May 25 '12 at 8:00
  • $\begingroup$ Do you have a link? I cannot find it with the search. $\endgroup$ – Potato May 25 '12 at 8:00
  • $\begingroup$ @joriki: You beat me to it. $\endgroup$ – Brian M. Scott May 25 '12 at 8:02