Last lecture we defined what it means that a polynomial with coefficients in a field is solvable by radicals over . Namely, there is a tower of extensions
where for each there is a positive integer such that and such that
[Friday’s lecture was covered by Marion Scheepers. Many thanks!]
Let’s return to the problem of solving quartic polynomials. In the first lecture on this topic, we reduced the problem of solving an equation like
to solving the similar problem
in which the coefficient of is zero. This is achieved by a simple translation, simply set . This was motivated and explained in that lecture. Let us now see how we can approach this problem.
1. We all know how to solve a linear equation such as , namely (assuming ; if then either and any is a solution, or else there are no solutions). This was known to Babylonian and Persian mathematicians (with the usual caveats about the signs of , since the notion of negative numbers had not been introduced yet.)
This is trivial, but there is a subtle point here:
If we are interested in solving polynomial equations in general, at some point we will need an argument justifying that we can. For now, let us proceed formally, assuming that we will always find solutions.
Just as with linear equations, we all know as well how to solve quadratics, such as . Namely, we can factor out (if we are in the linear case, so let’s assume that this is not the case) and then complete the square. We get
so iff , or
the well known quadratic formula.
Another small subtlety appears here, namely, there is some inherent ambiguity in the meaning of the expression . We usually resolve this by “choosing a sign” of the square root. As long as we are looking at quadratic polynomials with integer (or rational, or real, or even complex) coefficients, there is a standard way of making this choice. In more general situations (in arbitrary fields) there is no such standard procedure.
Besides this subtlety, a more serious one needs to be faced. Nowadays, we are used to working with complex numbers, so the view of a square root of a negative number does not cause confusion, but this was a serious issue for many centuries, and when complex numbers were first used, many were skeptical of whether they actually made sense. It wasn’t until Gauss’ presentation of complex numbers as pairs of reals that their use became mainstream. This is related to the question of whether one can always solve an equation. The answer was “no” until complex numbers were introduced and accepted, and then it became “yes.”
2. Cubic equations. The history of the solutions to cubic and quartic equations is full of melodrama, the historical note at the end of Chapter 2 of the book is well worth reading, see also this.