Why no general solution to quintic equations?

  • Thread starter Superhoben
  • Start date
  • #1
It seems odd to me that we have no general solution to quintic equations yet.
Is it possible that it exist any general solutions to equations of the fifth degree (and higher) that just haven't been discovered yet?
Or are we certain that it doesn't exist?
 

Answers and Replies

  • #2
Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
3,991
210
It is in fact a theorem that no solution exists that only uses functions that we would consider typical.

http://en.wikipedia.org/wiki/Abel–Ruffini_theorem

The theorem says that you cannot solve the general fifth degree polynomial using only basic arithmetic operations and calculating nth roots of numbers. It might be that there are other functions which you can allow yourself to use to solve higher degree polynomials (a trivial example is that if you define a function P(a,b,c,d,e) to return a root for a polynomial whose coefficients are a,b,c,d and e then you have "solved" the polynomial).
 
  • Like
Likes 1 person
  • #3
Thanks. I don't understand why it don't exist yet and I don't understand the proof completely. Just out of curiosity, how much math do you need to read to understand the proof?
 
  • #4
798
34
Thanks. I don't understand why it don't exist yet
It exists yet: The roots of the quintic equation can be analytically expressed thanks to the Jacobi theta functions, but not with a finie number of elementary functions.
 
  • #5
jbunniii
Science Advisor
Homework Helper
Insights Author
Gold Member
3,393
181
Thanks. I don't understand why it don't exist yet and I don't understand the proof completely. Just out of curiosity, how much math do you need to read to understand the proof?
A fairly large portion of a typical undergraduate course in abstract algebra. For example, the proof using Galois theory is given in the final chapter (33) of Pinter's A Book of Abstract Algebra, and it depends to some degree on almost all of the first 32 chapters.

There are more direct proofs that do not use Galois theory - indeed, Abel's original proof predated Galois theory. Here is a video which sketches one such proof:

 
Last edited by a moderator:
  • Like
Likes 1 person
  • #6
SteamKing
Staff Emeritus
Science Advisor
Homework Helper
12,796
1,668
There's all sorts of unsolved and insoluble problems in mathematics. It's not like walking into a restaurant and ordering a meal.
 

Related Threads on Why no general solution to quintic equations?

Replies
4
Views
639
Replies
3
Views
2K
Replies
2
Views
657
Replies
2
Views
3K
Replies
3
Views
4K
Replies
3
Views
18K
Replies
1
Views
1K
  • Last Post
Replies
15
Views
3K
Top