1. ## Galois group

Find the roots of f(x)=x^4+x^2+1. Find the Galois group of f(x).
I know that I need to use the resolvent cubic, but I don’t know how to find it and use it. Can someone explain it to me? Thanks.

2. Originally Posted by apple2009
Find the roots of f(x)=x^4+x^2+1. Find the Galois group of f(x).
I know that I need to use the resolvent cubic, but I don’t know how to find it and use it. Can someone explain it to me? Thanks.
$x^4+x^2+1=(x^2+x+1)(x^2-x+1)$

Can you take it from here?

3. Originally Posted by apple2009
Find the roots of f(x)=x^4+x^2+1. Find the Galois group of f(x).
I know that I need to use the resolvent cubic, but I don’t know how to find it and use it. Can someone explain it to me? Thanks.
Let a_3=0, a_2=1, a_1=0, and a_0=1. Plug them into the polynomial for a resolvent cubic (see here)

If my computation is correct, the resolvent cubic of f(x) is x^3-x^2-4x+4.

Then check how the resolvent cubic of f(x) factors and find the corresponding galois group (see here)