It turns out that Vieta's formula applies to any algebraically closed field. Call a field algebraically closed if(f) every polynomial in splits over .
Theorem: is algebraically closed.
Proof: It suffices to show that there is not a proper finite field extension of . This is a healthy exercise in Galois theory, and is thus left to the reader. (Hint: Suppose, by way of contradiction, that is a proper finite extension of . What can we say about ?)
Thus, we see that Vieta's formula works in at least one algebraically closed field. Let's broaden the scope.
where is an algebraically closed field, and that splits as
Proof: A field is necessarily commutative and distributes over addition. Thus, the proof is simply a matter of noting the coefficients must match up.
Thus, we can use Vieta's formula for arbitrary polynomials over fields.
[Edit: Sorry for the awful formatting. I'm used to writing things in pure LaTeX, so this is a little weird to me.]