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
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.