Has it been proven that quintic equations cannot be solved by *any* formula?

  • I
  • Thread starter swampwiz
  • Start date
  • #1
317
13
It seems that Abel's theorem says that the quintic cannot be solved by arithmetic & root operations, but couldn't there be the situation where another function is used in concert with these operations?
 

Answers and Replies

  • #3
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,728
It seems that Abel's theorem says that the quintic cannot be solved by arithmetic & root operations, but couldn't there be the situation where another function is used in concert with these operations?
If you restrict yourself to finite expressions, yes, that is true. However, if you allow such things as infinite series and the like, you can solve quintic equations---in terms of hypergeometric functions. See, eg.,
http://mathworld.wolfram.com/QuinticEquation.html
 
  • #5
317
13
OK, so it looks like the Bring radical could work, but it's an infinite series.
 
Last edited:
  • #6
Math_QED
Science Advisor
Homework Helper
2019 Award
1,701
720
It seems that Abel's theorem says that the quintic cannot be solved by arithmetic & root operations, but couldn't there be the situation where another function is used in concert with these operations?
Through Galois theory, it is proven that the general solution of a polynomial of degree at least 5 is not expressible in terms of the operations addition, multiplication (and their inverses) and taking roots.

Other ways are still possible.
 
  • Like
Likes jedishrfu
  • #8
12,077
5,745
Historically, mathematicians would compete for solving various types of polynomials. Tartaglia, an amateur Italian mathematician came up with the general formula for roots of a cubic and created a poem encoding the formula to prevent others from claiming they found it first.

https://www.storyofmathematics.com/16th_tartaglia.html

From there Ferrari, another younger mathematician conquered the quartics and the quintics remained unsolvable until Abel definitively proved they were by Galois theory.
 
  • #9
WWGD
Science Advisor
Gold Member
2019 Award
5,349
3,327
It depends on whether the associated Galois group of the roots is a solvable group.
 
  • Like
Likes Math_QED
  • #10
WWGD
Science Advisor
Gold Member
2019 Award
5,349
3,327
And we know both that, of course there are nonsolvable groups and that these are the Galois groups of some polynomials.
 
  • #11
mathwonk
Science Advisor
Homework Helper
11,036
1,229
of course "roots" are also given by infinite series.
 

Related Threads on Has it been proven that quintic equations cannot be solved by *any* formula?

Replies
2
Views
650
Replies
3
Views
4K
Replies
6
Views
5K
Replies
3
Views
2K
Replies
4
Views
626
Replies
5
Views
3K
Replies
7
Views
841
Replies
4
Views
2K
Replies
66
Views
12K
Replies
4
Views
2K
Top