Galois group

Oct 2009
56
0
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.
 

chiph588@

MHF Hall of Honor
Sep 2008
1,163
429
Champaign, Illinois
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.
\(\displaystyle x^4+x^2+1=(x^2+x+1)(x^2-x+1) \)

Can you take it from here?
 
May 2010
95
38
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)