TITLE: Prove cubic equation on $\mathbb{C}$ can be expressed as a product of linear factors
QUESTION [1 upvotes]: I want to ask about Exercise 2.8 in Ian Stewart's Galois Theory states that: "Without using Fundamental Theorem of Algebra, prove that a cubic polynomial on $\mathbb{C}$ can be expressed as a product of linear factors".
Exercise 2.7 states a similar question but we consider the cubic polynomial on $\mathbb{R}$. I solved it by noting that it must have a root in $\mathbb{R}$, and any quadratic polynomial with coefficients in $\mathbb{R}$ can be expressed as a product of linear factors. 
Obviously, the same idea can't be applied in Exercise 2.8. Help me with this

REPLY [1 votes]: I'm guessing you can still use the fact that cube roots and square roots exist in $\mathbb{C}$. Then given a cubic, you can apply Cardano's method (or just a cubic formula https://en.wikipedia.org/wiki/Cubic_function). This will give you one root, say $z_1$. Then you can factor your polynomial as follows:
\begin{equation*}
f(x)=(x-z_1)g(x)
\end{equation*}
where $g(x)$ is a quadratic. Then apply the quadratic formula to $g$ to get the remaining factors.
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