Remember the quadratic formula? It looks something like this:

$fraction(-b+square_root(b^2-4ac), 2a)$

But that's just some expression. If you think a little harder, you might remember its immediate context and meaning: If you have three known quantites $a, b, c$ and some unknown $x$, but you know that $ax^2 + bx + c = 0$, then you can deduce the value of $x$--it is the expression you see above.

But when does that happen? Mathematics is always concerned with "finding $x$" almost as a game in its own right, but really the quest for $x$ most often starts with a real problem. Maybe we want to predict the motion of a canonball. We intuitively know the arc through the air resenbles a parabola, and maybe even remember that the quadratic equation above also looks like a parabola. So if we know some physics maybe we can model the canonball with a quadratic equation, whereupon this formula will let us deduce things about it's trajectory.

We start with a real-world problem, model it as a mathematical one, use mathematics to solve the model, and then interpret the results as predictions about the real world. In the case of the canonball, the mathematical model is an equation, and the mathematical technology that takes us through the bottom half of that picture is the quadratic formula.

This is not uncommon: An equation simply captures the numerical relationships between numbers. If we can solve equations, that simply means that we can deduce numerical consequences from these relationships. Scientific laws are often about relationships between measurable quanities--that is, equations. Applying scientific laws, then, is about deducing the consequences of those relationships in various specific situations. That is, solving equations.

The lowly quadratic formula makes a very brash claim, then: Any time you have a relationship between numbers that is quadratic, a single formula will furnish all the consequences of that relationship without exception. When we happen on something so powerful and striking, it is valuable to ask how this came about and how far we can push that mechanism to get even more powerful results.

This document will be a journey of thinking deeply about simple things. Broadly, the direction of this journey will be toward the generalisation of the quadratic formula to cubic equations: $ax^3 + bx^2 + cx + d$. However, the real goal is an effective understanding of equations of this kind, and the technology that helps to solve them not just numerically, but in a way that gives us a precise understanding of how those solutions depend on the situation we were given and change with it.

We will gently but unapologetically introduce abstractions from the field of abstract algebra when useful, and will equally unapologetically gloss over others. By the end, you will have exposure to high level ideas from group theory, discrete harmonic analysis, and Galois theory. Those names, however, will still more or less be just words, as that's all they ever really are.

Our goal is to solve the cubic equation $x^3 + bx^2 + cx + d = 0$ with rational coefficients. We will start by thinking about a simpler case: the quadratic formula. $x^2 + bx + c = 0$. Equivalently, if $x^2 + bx + c = (x-alpha()_1)(x-alpha()_2)$, we want an explicit formula for $alpha()_1$ and $alpha()_2$ in terms of $b$ and $c$ that uses only elementary operations (arithmetic operations and taking roots).

### Chapter $x^2 + 1$: Complex Conjugation

Let's start with an example: $x^2 + 1 = 0$. We know the solutions to this: $i$ and $-i$. In fact, we know a relationship between these two solutions: They are complex conjugates of another, which we will denote with a $tau$: $-i = tau(i)$. This is not a coincidence, but a result of the following observation: If we have a complex number $alpha()$ that is a root of this equation: $alpha()^2 + 1 = 0$, then we can take complex conjugates of both sides:

$tau(alpha()^2 + 1) = tau(0)$

Certainly $tau(0) = 0$, so the right side is easy to simplify. But the left side also simplifies: The important point is that with complex conjugation, it doesn't matter whether you first add two numbers and then conjugate them (like $tau(a+b)$) or conjugate them first and then add them (like $tau(a)+tau(b)$)--the result is the same: $tau(a+b) = tau(a)+tau(b)$. And similarly for multiplication.

This property is really quite rare in the world of mathematics: $square_root(1) + square_root(4)$ is 3, and not $square_root(5)$, $sin(pi())+sin(pi())$ is -2, and not $sin(2pi())$ (which is 0), and $1^2 + 2^2 = 5$ and not $3^2$. This property is also pretty rare in the real world: If you put on your socks and then your shoes, the result will be quite different to if you put on your shoes first and then your socks! When an operation can be done before or after both addition and multiplication with the same result, it is therefore quite special. Such operators are called "homomorphisms" (literally "same shape").

The fact that $tau()$ is a homomorphism allows us to rewrite the equation as

$tau(alpha())^2 + 1 = 0$

That is, $tau(alpha())$ is also a solution to $x^2 + 1 = 0$! This is a very general fact: If we have any polynomial $f(x)$ with rational coefficients, then $tau(f(alpha())) = f(tau(alpha()))$, so if $2+i$ is a root of $f(x)$, then you know immediately that so too is $2-i$.

### Chapter $x^2 - 2$: More general conjugation

Now let's take a different example: $x^2 - 2 = 0$. Any elementary student will tell you the solutions to this are $square_root(2)$ and $-square_root(2)$. However

We start with the $alpha()_1 = sig(alpha())$

### Chapter $(x^2+1)(x^2 - 2)$: Groups of symmetries

### Chapter $x^3-2$: Cubic symmetries

### Chapter $(x+1)^3-2$: Fields are better than polynomials