This is inspired by the review on "Linear Algebra done right". I decided to do a top-level post, because it hits a misconception that is pretty common.

The starting point of this post is this quote from "Linear Algebra done right":

Remarkably, mathematicians have proved that no formula exists for the zeros of polynomials of degree 5 or higher. But computers and calculators can use clever numerical methods to find good approximations to the zeros of any polynomial, even when exact zeros cannot be found.

For example, no one will ever be able to give an exact formula for a zero of the polynomial p defined by $p (x) = x^{5} - 5 x^{4} - 6 x^{3} + 17 x^{2} + 4^{x} - 7$ .

The authors misrepresent an important point that is understood by most mathematicians, but not properly understood by many laypeople.

What does it mean to solve a problem? What does it mean to have an exact formula for the solution of a problem?

The answers to both are a social convention that has historically changed and is expected to continue to evolve in the future.

Back in the days, people only considered rational numbers, ie fractions. Oh, but what about the positive solution to $x^{2} = 2$ ? Ok, we can't express this as a rational number (important theorem). Because these kinds of problems occured quite often, the mathematical community arrived at the consensus that $\sqrt{2}$ , or more generally $\sqrt{r}$ for nonnegative $r$ should be considered an explicit solution. Amazingly, this allows us to express the solution to any quadratic equation $0 = a x^{2} + b x + c$ explicitly, with our expanded notion of "explicit". From an algebraic viewpoint it was natural to bless the positive solution to $x^{n} = a$ as an "explicit formula" next; historically it was a more contentious thing, because greek geometry wanted numbers to be constructible using a ruler and compass only. "Doubling the cube", ie expressing the positive solution to $x^{3} = 2$ as a geometric construction was a famous old problem (proven impossible in 1837, after having been a very prominent mathematical research problems for more than 2000 years).

Now, this obviously says not a lot about the cube root of 2, but says a lot about "constructible with ruler and compass".

In other words: "Explicit solutions" are a messy historical map to mathematical territory, nothing more.

The same holds if you ask for explicit formulas for zeros of polynomials after...

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