There’s the quadratic formula and the one for 3 and 4 factor ones (apologies if I’m wording this wrong), but I just heard that apparently there isn’t anything like a quintic(?) formula and so on. Why is this?? Googling gives me a bunch of confusing terminology that’s difficult to parse.
In: 75
The (most common) proof is way, WAY beyond ELI5 level. It occupied about half a semester of graduate-level abstract algebra at my university, and the progression of the proof is not at all obvious, or at least wasn’t at all obvious to me (and I am pretty good at this sort of thing). It’s a pretty extraordinary piece of mathematics, all the more so for having been done by one guy at age 18.
But I’ll sketch it out, broadly speaking.
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So. We need to have some way of describing a mathematical object that tells us whether a polynomial has a solution. It’s not even obvious where we would *start* with this. But here’s an idea: what if we think about what would happen if we had the rational numbers and we “attached” the roots of the polynomial to it.
So for example, if we have the polynomial x^2 + 1 = 0, we can “attach” +i and -i (the two roots of this polynomial) to the rational numbers. This turns out to get us all numbers of the form a + bi where a and b are both *rational* numbers (note that this is a subset of the complex numbers, not all of them). We call this operation a **field extension**, because both objects are fields, a kind of mathematical object that “acts like” the rational numbers in some sense. (Specifically, a field is a set on which you can add, subtract, multiply, and divide by non-zero values, and it turns out that “take the rationals and stick the roots of a polynomial on them” always results in such an object.)
More abstractly, we take the field Q of rational numbers (Q is the usual symbol for them), and for any polynomial P with roots x1, x2, x3, …, xn we can create a new field which we write Q[x1, x2, x3…, xn] of the rational numbers with these extra roots. We call this new, bigger, field the splitting field of our polynomial P.
This is helpful because it takes us out of the realm of our polynomial, and into the realm of talking about abstract algebraic objects (which is usually where mathematicians like to live). But how the hell does this help us?
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Well, it turns out – and again this is not at all obvious and takes a lot of work to prove – that the relationship between Q and the splitting field Q[x1, x2, …, xn] encodes the information we want.
Return to our earlier example: we took the rationals and added +i and -i to them. But the properties of these “rational complex” numbers wouldn’t change if we swapped +i and -i, and we wouldn’t be messing with the rationals that lack any imaginary part by doing so. In other words, we have an operation that:
* Preserves the properties of the bigger field (the splitting field) *and*
* Does not change the smaller field, in our case Q, at all.
If we take all the operations that do this – and there may be quite a few of them – they form another kind of mathematical structure called a *group*. Groups are a very common type of mathematical object, because they in some sense describe symmetries and transformations in a very general way, and studying the symmetries of an object is often a way to understand its properties.
This particular group, which we call the *Galois group*, encodes information about how our polynomial’s roots extend the rational numbers. In some cases, when the polynomial’s roots are all rational themselves, it doesn’t extend the rationals at all (because you could already “get to” those numbers). In other cases, it extends the rationals in various ways.
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Okay, but how does *that* help us?
Well, it turns out that the properties of the Galois group tell us something about the original polynomial’s roots – namely, if they can be described using just arithmetic and nth root operations.
It turns out that roots that can be described this way extend the rationals in a very specific kind of way. The resulting extensions – and their corresponding Galois groups from the previous section – can only take on a particular kind of structure.
Since we can build up our full extension by *all* the roots by extending by each one one at a time, we get a sequence of Galois groups for each of those extensions in turn. And it turns out that if that sequence has particular properties – which turn out to be equivalent to the full Galois group being something called a [solvable group](https://en.wikipedia.org/wiki/Solvable_group), the original polynomial had a solution that could be written using only arithmetic and radicals.
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Finally, we show that there exists at least one polynomial of degree 5 – it turns out that x^5 – x – 1 works – whose Galois group is *not* solvable. Then we work backward:
* This polynomial has a non-solvable Galois group.
* Therefore, its chain of extensions does not have the property that it would have if every root could be written with only arithmetic and radials.
* Therefore, **this** polynomial has no solution with only arithmetic and radicals
* Therefore, there is no general formula that can solve every polynomial that way.
It turns out that the smallest non-solvable Galois group requires at least five roots, and therefore a polynomial of degree at least 5, which explains why degree five can’t be solved with a formula using only arithmetic and roots (and 2, 3, and 4 can be).
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