Is there a simple explanation why degree 5 polynomials (and up) are unsolvable?

Question

We can solve (get some kind of answer) equations like:

$$ ax^2 + bx + c=0$$

$$ax^3 + bx^2 + cx + d=0$$

$$ax^4 + bx^3 + cx^2 + dx + e=0$$

But why is there no formula for an equation like $$ax^5 + bx^4 + cx^3 + dx^2 + ex + f=0$$

I'm not sure if this has anything to do with the Galois theory, but is there a dumbed-down simple explanation as to why degree 5 polynomials (and up) are unsolvable?

Answer 1

I'll try a "dumbed down" version, although @Robert Israel's answer plus comments are fine!

Solvable means solvable by radicals, and that means that, starting from the polynomial equation, you can only do 1) field arithmetic $(+,-,\times,\div)$, or 2) "extracting roots; e.g. square roots, cube roots, etc. It is the case, by Abel-Ruffini first and then by Galois, that there is no general "formula" for solving polynomials above degree 4. Naively, that suggests that the formula gets "too complicated" at some point. @paul garrett gets at this when he refers to the resolvent, which is a step that can simplify solving if the resolvent polynomial is of lower degree.

Galois found that the way to measure "too complicated" is by checking which roots of the polynomial can be "switched around", or permuted, while maintaining certain equations of the roots. For example, if you are working over the rational numbers, then you can't switch around any rational number without changing important relationships. That seems obvious. But what might seem strange is that for a polynomial like $x^2-2$, whose roots are $\sqrt2$ and $-\sqrt2$ , you can switch these around and not hurt any other arithmetic!

The way to formalize what it means to "switch around" roots is thought group theory, and there is a group that corresponds to how the roots of a polynomial can be switched around called the Galois group. Finally, if this group is "too complicated" (i.e. too many ways to permute the roots), then that group and its corresponding polynomial are not solvable by radicals. In the case of 5th degree polynomials, if it were possible to "invert" the polynomial $x^5-x-1$ (i.e. solve it directly like we can $x^5$), I believe this is all that would we needed for all 5th degree polynomials to be solvable by radicals. So as you see, it's just a "little bit" too complicated, and it gets worse as the degree increases.

I'm leaving out lots of details, but the other answers and links fill in those details. But I hope this gives you a flavor if what's going on.

Answer 2

It has everything to do with Galois theory, although the original proof preceded Galois.

See this Wikipedia article on Ruffini theorem.

I don't believe there is a "dumbed-down simple explanation".

Answer 3

There is already one "dumbed" answer above, so forgive me for adding an even "dumber".

What it means by solvable is to be able to write down a formula for a solution of a polynomial equation like $ax^2 + bx + c = 0$ such this one:

${x = \frac{{ - b \pm \sqrt {b^2 - 4ac} }}{{2a}}}$

In other words, we need the solution (whatever it would be in $x$) could be expressed in some algebraic expressions. In the example above: $-, b, \pm, \sqrt, \: ^2, 4, a, c, /,$ and $2$ are all algebraic expressions.

For most of all numbers we know, they could be expressed in algebraic expressions. For example: $2, 6/3, \sqrt{2}, \sqrt{-1},$ etc and even for a non-repeating irrational number like $\pi$ could be expressed in $\frac{C}{D}$ where $C$ is the circumference of a circle and $D$ is the diameter.

Amazingly there are numbers that could not be expressed in any algebraic expressions we know.

Historically, it was discovered when observing that polynomial equations of degree higher than 4 may not necessarily have a solution that could be expressed in algebraic expressions. It was first observed by Joseph-Louis Lagrange in 1770, partly proven by Paolo Ruffini in 1799, and then completed by Niels_Henrik_Abel in 1824, establishing Abel–Ruffini theorem. But not until 1830 Évariste Galois at the very young age of 18 showed the necessary condition of a polynomial equation to have a solution hence called solvable.

The theory is called Galois theory, but unfortunately it is too complex to be explained satisfactorily here. If we have enough background in fundamental algebra, it would still need at least 200 hundreds pages of hard math to understand it. Nevertheless the necessary condition showed by Galois theory also shows that there are polynomial equations that their solutions could not be expressed in any algebraic expressions.

An example of a such unsolvable polynomial equation is $x^5 - x - 1 = 0$. Note to not be confused with the algebraic expression mentioned above. In the equation $x^5 - x - 1 = 0$, we cannot transform it into a solution expression where a single $x$ is put on the left side of the equation such $x = \: ...$ because, well, we don't have the algebraic expression to be written down in the right side of the equation.

Of course we could approximate the solution with some number close enough, but it is still NOT the solution to the equation because simply it is just close enough BUT not the exact solution. The exact solution is not expressible yet. Amazing right!