# Help Understanding the paragraph in page 16 of Humphreys Lie Algebras

The following paragraph is in page 16 of "Introduction to Lie Algebras and Representation Theory - Humphreys"

$$K$$ is a subalgebra of $$\mathfrak gl$$($$V$$). Denote $$W$$={$$w\in V$$ : $$x.w=\lambda(x)w$$, for all $$x \in K$$}, $$i.e.$$ the set of common eigenvectors of $$K$$. Let $$L=K+Fz$$ (for any $$z\in L\setminus K$$) and [use the fact that $$F$$ is algebraically closed to find and eigenvector $$v_0\in W$$ of $$z$$.] Then $$v_0$$ is obviously a common eigenvector for $$L$$.

I can't understand the part [ ]. I mean, I don't know how to use the algebraically closedenss of $$F$$. I guess it's quite easy part but I'm stucked. Thanks for any help in advance.

If $$F$$ is an algebraically closed field and $$V$$ is a finite-dimensional vector spae, then every linear map $$L\colon V\longrightarrow V$$ has some eigenvalue, since the eigenvalues are the roots of the characteristic polynomial of $$L$$. If $$F$$ is not algebriacally cloed, then $$L$$ may not have eigenvalues.