1. ## Galois Theory

How can I find the Galois group of x^5 + 2x +2?

2. Originally Posted by peteryellow
How can I find the Galois group of x^5 + 2x +2?
If you reduce mod $\displaystyle 29$ the polynomial factors as $\displaystyle (x^2+4x+7)(x^3+25x^2+9x+21)$. Sorry if this method is really computational, this is the only method I was able to think of.

Therefore by Dedekind's theorem the Galois group contains a product of a 2-cycle and 3-cycle. The only transitive subgroup of $\displaystyle S_5$ which contains a product of a 2-cycle and a 3-cycle is $\displaystyle S_5$ itself. Therefore, the Galois group is $\displaystyle S_5$.

3. Thanks, but I have a question why are you reducing modulo 29 why not any other number?

I have also seen this method somewhere else, but I couldnot see why they choose these numbers as you have choosen 29?

4. Originally Posted by peteryellow
Thanks, but I have a question why are you reducing modulo 29 why not any other number?

I have also seen this method somewhere else, but I couldnot see why they choose these numbers as you have choosen 29?
I started from p=3 and worked myself up to p=29 until I got a the result I wanted.
I used computer software for this, I did not spend all my time doing this by hand.