Solve with radicals the following equation $x^3+x+3=0$, using Galois Theory.
I'm just starting learning this and I do not have many ideas.
Asked
Active
Viewed 174 times
1
Empiricist
- 8,141
- 1
- 25
- 42
Weijie Chen
- 801
-
1I tried to describe the process in this answer. I'm sure many textbooks (N. Jacobson, Basic Algebra I, would be my recommendation) do it better and more verbosely. – Jyrki Lahtonen Jul 30 '15 at 08:19
1 Answers
0
The abel ruffini theorem in galois theory is not a tautology its just an implication which leads us to a contradiction. In simple terms if a polynomial is solvable by radicals then a normal extension of fields must exist and since there are some quintic equations which cannot contain a ladder of normal extensions hence we arrive at the conclusion that not all polynomials are solvable by radicals. But at no point does Galois theory promise you the existence of roots in radical if a ladder of normal extensions exists. it is not upto Galois theory to come up with such an equation(s) in radicals. The solution to the above problem will not give you any insight into how radicals are directly related to the normal extensions.But it will certainly give you a good picture about the ladder of normal extensions.That is all you would need to understand the the fundamental theorem on Galois theory.