# Deriving Formulae for the roots of the quartic and cubic polynomials

I have seen derivations of the general solution for the roots of fourth and third degree polynomials of 1 variable in Dummit & Foote's Abstract Algebra; however, it was by no means simple to me. I am curious to know if there is a more natural approach using techniques from Algebraic Geometry (i.e. using Groebner Bases and Elimination Theory).

# My Attempt (cubic):

Consider the equation $f(x)=x^3 +ax^2 +bx+c$. By the Fundamental Theorem of Algebra, there exists $x_1, x_2, x_3 \in \mathbb{C}$ so that $f(x) = (x-x_1)(x-x_2)(x-x_3)$. After multiplying everything out, we get the following system of polynomial equations by comparing coefficients.

\begin{cases} a=-x_1-x_2-x_3 \\ b = x_1 x_2 +x_1x_3+x_2x_3 \\ c= -x_1x_2x_3 \end{cases}

If we consider the ideal, $I = \langle x_1+x_2+x_3 +a, \;x_1x_2+x_1x_3+x_2x_3 - b,\; x_1x_2x_3 + c\rangle \subseteq \mathbb{C}[x_1,x_2,x_3]$. My idea is to find a Groebner basis using lexicographic order ($x_1 > x_2 > x_3$) which will hopefully lead to a simpler expression which can be solved. After putting the ideal into Macaulay 2, I arrived with the following basis for $I$.

$$\mathbb{G} = \{x_1+x_2+x_3 +a, \; x_2^2 + x_2x_3+x_3^2+ax_2+ax_3 +b, \; x_3^3 +ax_3^2 + bx_3 + c \}$$

However, this method doesn't seem to be leading anywhere since the final equation is exactly what we started with.

My question: can this method can be modified so that it actually works, and if so, can it be extended for the quartic?

• Try getting rid of the $x$ term of the $x^2$ term. You can do this some substitution like $x=y-d$. If you make this substitution, for what value of $d$ does the coefficient of $x^2$ become $0$? For what value of $d$ does the coefficient of $x$ become $0$?
• Once you have gotten rid of one of the terms, you need to somehow reduce it to a linear or quadratic equation. However, you have an exponent of $3$ in this equation, so you'll likely need to make a linear or quadratic equation in terms of $x^3$. What kind of substitution do you think will lead to this?