I've been trying to prove the impossibility of the quintic "on the cheap" without having to go through a graduate course in abstract algebra (I haven't even done the undergraduate course, although I've been reading up on it a little bit at a time). I understand Bezout's Lemma, with a practical example being the way that it is possible to tighten all the lug nets on a wheel by incrementing by the same angle, without repeating any, so long as the increment in terms of lugs and the total # of lugs have a Greatest Common Divisor of 1 (there is a better term for this, but it has slipped my mind at the moment).

It seems that if the quintic is impossible, then since a quintic is a special case of a hexic, heptic, etc., then those must be impossible too. Also, there is no point in proving that a quartic is possible since a formula already exists for it. Therefore, the full brunt can be directed at the quintic.

As for the quintic, I was reading this

https://www.quora.com/Why-is-there-no-formula-for-solving-polynomials-with-n-4-explain-the-answer-in-such-a-way-that-a-calculus-1-or-precalculus-student-can-understand-if-possible

which goes into explaining that there can be functions expressed in terms of roots of (A,B,C,D) for degree n and (p,q,r) for degree n-1.

n = 3:

n = 4:

such that no matter how the roots for (A,B,C,D) are swapped around, the resultant (p,q,r) are the same. And for a formula to exist, every root must be swappable with any other root (although I am not so clear on what this exactly means).

However, it is impossible to come up with any function in (A,B,C,D,E) in which the all the permutations of swapping work, and thus it seems that it could be proven through exhaustion that this is the case, although I am not sure what these functions would look like.

I'd appreciate any insight that will help me understand this "on the cheap".

It seems that if the quintic is impossible, then since a quintic is a special case of a hexic, heptic, etc., then those must be impossible too. Also, there is no point in proving that a quartic is possible since a formula already exists for it. Therefore, the full brunt can be directed at the quintic.

As for the quintic, I was reading this

https://www.quora.com/Why-is-there-no-formula-for-solving-polynomials-with-n-4-explain-the-answer-in-such-a-way-that-a-calculus-1-or-precalculus-student-can-understand-if-possible

which goes into explaining that there can be functions expressed in terms of roots of (A,B,C,D) for degree n and (p,q,r) for degree n-1.

n = 3:

**AAB + BBC + CCA = p****ABB + BCC + CAA = q**

n = 4:

**AB + CD = p****AC + BD = q****AD + BC = r**

such that no matter how the roots for (A,B,C,D) are swapped around, the resultant (p,q,r) are the same. And for a formula to exist, every root must be swappable with any other root (although I am not so clear on what this exactly means).

However, it is impossible to come up with any function in (A,B,C,D,E) in which the all the permutations of swapping work, and thus it seems that it could be proven through exhaustion that this is the case, although I am not sure what these functions would look like.

I'd appreciate any insight that will help me understand this "on the cheap".

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