Linear Algebra - direct sums

Question

Let ${V}=\mathbb{R}^3$, and define

${V}_1=\{{\textbf{x}\in{V}\mid x_1+x_2+x_3=0}$}, and ${V}_2=\{{\textbf{x}\in{V}\mid x_1=x_2=x_3}$}.

We know that $V$ is a direct sum of $V_1$ and $V_2$. Can we find an example by replacing $\mathbb{R}$ by some other field $\mathbb{F}$ such that $V$ fails to be the direct sum?

So we need to find a field such that at least one vector in the new field cannot be written as a sum of $V_1$ and $V_2$. I am stuck on this... Can someone give me a hint?

Answer 1

If $\mathbb F=\mathbb Z_3$, then they fail to be a direct sum, since $(1,1,1)$ belongs to both of them.