alt Hacker NewsI don't know if I read this right, but I thought it's proven that "elementary functions" can't solve 5th degree or higher polynomial, so I'm confused how it's interpreted if elementary functions also include arbitrary polynomial roots. Or is it different elementary functions?
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The term 'elementary function' doesn't really have a single universally agreed on strict definition.
Definitions are either a bit fuzzy, or not universally agreed on.
Though interestingly https://en.wikipedia.org/wiki/Elementary_function says "More generally, in modern mathematics, elementary functions comprise the set of [...]". Though at least Wikipedia thinks that 'modern mathematics' has a consensus; of course, there's no guarantee that whoever you are talking to uses the 'modern mathematics' definition that Wikipedia brings up.
That theorem is not formulated about "elementary functions".
It says that polynomial equations of the 5th degrees or higher cannot, in general, be solved using "radicals".
While something like "polynomials" or "radicals" has a clear meaning, which are the "elementary functions" is a matter of convention.
The usual convention is to include all algebraic functions and a few selected transcendental functions.
In "all algebraic functions", are included the rational functions, the radicals and the functions that compute solutions of arbitrary polynomial equations.
Some conventions used for "elementary functions" describe the expressions that you can use to write such "elementary functions", in which case not all algebraic functions are included, but only those written by combining rational functions with radicals.
For an algebraic function that computes a solution of a general polynomial equation, which cannot be expressed with radicals, you cannot write an explicit formula, but you can write the function only implicitly, by writing the corresponding polynomial equation.
So the difference between the 2 kinds of conventions about which are "the elementary functions" is usually based on whether only explicitly-written functions are considered, or also implicit functions.