Would an extremely advanced civilization be able to find the exact roots of polynomials higher than degree 5?

how is sqrt(2) an exact root?
en.wikipedia.org/wiki/Bring_radical

sqrt(2) is an exact number

>numerical approximations != exact roots
too complex to be used in practice != impossible

look at the page I linked, will you? BR(a) is an exact number as well

Of course you can use them in practice you brainlet.

It still doesn't mean you can have a machine where you plug in 5 or more different coefficients for a polynomial to the fifth or higher degree and have it spit out all of the exact roots in every case.

Being used in practice is not the same thing as having an exact value.

>It still doesn't mean you can have a machine where you plug in 5 or more different coefficients for a polynomial to the fifth or higher degree and have it spit out all of the exact roots in every case.
Actually that's exactly what it means when you have 5 coefficients. What did you think 'algebraic solution to the general quintic' means?

what kind of machine? a normal computer can't write sqrt(2) as a number either. if you mean representing solutions adding other things, yes you can do it for degree 5 with

What about nth degree? Whats stopping a sufficient logic device from spitting out exact values to every root for every case of a polynomial to any degree N higher than 5

I dunno, I'd guess you can definitely do it for a fixed N

The reals are not enumerable so a machine could never spit out exact roots without an algebraic algorithm.

wait so the roots would have algebraic algorithms with it in order to represent the roots?

nonsense. it has nothing to do with that. the roots to a polynomial are in a finite extension of Q[a1,...,an] where ai are the coefficients, which is enumerable

Wait I mean is regarding those proofs that it is impossible to derive a general formula for all cases of a polynomials function for degrees higher than 5. Couldn't undiscovered branches of mathematics disprove that since there would be new ways to represent that number?

And I guess are current "approximate roots represented by bounds/margin of error or whatever" would count as a representation of those roots, but those aren't exact x values such that I could plug it into f(x) and get exactly zero

writing sqrt(2) is as precise as defining some x to be the unique solution an algorithm is converging to

but sqrt(2) is a number I can plug into a f(x) polynomial and get zero.

Can I plug in a "value an algorithm is converging to" and get exactly zero?

how do you plug it in exactly? you write "sqrt(2)" in the board and eventually cancel it out because you know the rules it follows? you can do exactly that with x...

but those are only bounds

They are not exact roots.

you're really not following

if I have a polynomial that factors into +- sqrt(2) and I plug sqrt(2)=x into f(x) I will get exactly zero

If I plug in "numerical approximation"=x into f(x) I will get approximately x

I want to know if an advanced civilization could get "exact number"=x into f(x) and get exactly zero for any arbitrarily large polynomial

take a (good) algorithm A. take your polynomial P. define A(P) as the unique value the algorithm converges to. plug in A(P).

Would that be zero or something that converges to zero?

that would be zero. A(P) is a number.

Also why do I need to use an algorithm? A and the other roots should be able to exist somewhere on the number line, as such shouldn't I be able to point to it and be oh like this is number DERP. Like I can point to the sqrt(2) of be like this number is the sqrt(2). I can't point to a singular number if its bounds or an approximation

A(P) is a number. you can point at it in the real line. just like you can point at sqrt(2) with your finger because you have arbitrarily good bounds for where it is.

1. say I have a polynomial to the 11th degree and I find the roots of it. Are those numbers going to be less exact/precise/defined as the sqrt(2)

2. Do these methods hold up to degrees that are extremely high

>1. say I have a polynomial to the 11th degree and I find the roots of it. Are those numbers going to be less exact/precise/defined as the sqrt(2)
Depends on the polynomial. The roots of x^(11)=0 are arguably more precise.

>2. Do these methods hold up to degrees that are extremely high
Depends on the polynomial.

jesus christ shut the fuck up. this guy is confused enough without you idiots popping up every few posts writing intentionally misleading nonsense

>jesus christ shut the fuck up.
Why the vulgarity?

>depends on the polynomial

So its not true for all cases in that "all cases of all polynomials" can be factored down to its roots

>this guy is confused enough without you idiots popping up every few posts writing intentionally misleading nonsense
What's misleading about it?

>So its not true for all cases in that "all cases of all polynomials" can be factored down to its roots
All polynomials over [math] \mathbb{C} [/math] factor into linear factors over [math] \mathbb{C} [/math], see en.wikipedia.org/wiki/Fundamental_theorem_of_algebra#Corollaries

but instead of the root being sqrt(2) its "somenotation"(2)

but this notation is a bound no? If it is a bound that means it isnt a precise number but an amalgam of numbers close together

>but this notation is a bound no?
I don't know what notation you're referring to.

Not if solving it means solving it by radicals, because we have a theoretical argument stating that the generic equation in 5+ variables cannot be solved by radicals.
Now, as others have pointed out, it's only a matter of convention. There's no reason to believe that the equation is solved when we have an expression involving radicals or that we should limit ourselves to solving in radicals. The only reason is that x^n is bijective on R^+ and we know how to compute the n-th roots of a real/complex with arbitrary precision.
If we had a similar algorithm for extracting the roots of some other family of polynomials, we could add them to our toolkit of "equations we can solve" and see what our notion of solvability becomes.

>5+ variables
shit I meant in degree 5+

ok now tell us: are traps gay?