In this case it's the most straightforward way. If you're talking about computational complexity, then I believe some algorithms still factor the polynomial to compute the group, so I don't know if there's a way around it in general.
@@coconutmath4928 i mean why we use the galois group for example to solve a cubic equation when we can solve it easily using cardon's method which involve the real and imaginary solutions
The point of Galois theory is not to find the solutions of polynomials, but rather to find intermediate fields in a (finite) Galois extension. For example, in this case we know there is only one intermediate field ℚ ⊂ M ⊂ K, since the Galois group ℤ/4ℤ only has one proper nontrivial subgroup. I feel like a lot of expository material focuses a lot on the insolubility of the quintic. While this is an elegant application of the theory, thinking that is all kind of misses the point.
So interesting!
Hey bro, need some algebraic topology, moduler forms
Try to make video to find intermediate fields
👍
Hate the fact that to estimate the gailois group, one has to actually solve the polynomial. What is it worth then to calculate the group?
In this case it's the most straightforward way. If you're talking about computational complexity, then I believe some algorithms still factor the polynomial to compute the group, so I don't know if there's a way around it in general.
sir what is the point of the galois theory it can't give us exact or approximated solution
Can you clarify this? We did find the automorphisms of G explicitly.
@@coconutmath4928 i mean why we use the galois group for example to solve a cubic equation when we can solve it easily using cardon's method which involve the real and imaginary solutions
The point of Galois theory is not to find the solutions of polynomials, but rather to find intermediate fields in a (finite) Galois extension. For example, in this case we know there is only one intermediate field ℚ ⊂ M ⊂ K, since the Galois group ℤ/4ℤ only has one proper nontrivial subgroup.
I feel like a lot of expository material focuses a lot on the insolubility of the quintic. While this is an elegant application of the theory, thinking that is all kind of misses the point.