The argument we gave in class for the existence of eigenvectors for operators on finite dimensional complex vector spaces (and for the existence of invariant planes for operators on finite dimensional real vector spaces) uses the fundamental theorem of algebra. One can actually prove the existence of eigenvectors without appealing to this result, although the argument is more complicated.
As a corollary, one obtains a linear algebra proof of the fundamental theorem of algebra, which seems like a nice outcome.
The details can be found in a nice paper by Harm Derksen, currently available through his website or in JSTOR (American Mathematical Monthly, Vol. 110 (7) (2003), 620-623). A variation of the proof (perhaps more accessible) is in this paper by Keith Conrad, currently available through his website.
There is a slight disadvantage to both papers (which is perhaps the reason why I am not presenting their result in class) if we want to follow the approach of the textbook, and avoid introducing determinants at this stage. The problem is Corollary 4 in Conrad’s paper or Lemma 4 in Derksen’s, that operators on odd dimensional real vector spaces admit eigenvectors. Their proofs use determinants. The proof we gave (or are in the midst of giving) in lecture avoids determinants, but of course uses the fundamental theorem (so we can find an invariant plane and then argue by induction).
Can you find a way of obtaining this result without appealing to either determinants or the fundamental theorem, so we have a proof of the existence of eigenvectors compatible with the philosophy of the textbook and entirely self-contained?
(Note that an odd degree polynomial with real coefficients has a real root, and this can be proved very easily. From this, the argument for operators on does not require the fundamental theorem, and we can extend this to operators on , again avoiding the theorem, because we have explicit formulas that allow us to factor a quartic into the product of two quadratics. Can we find an argument for operators on ?)