.Determine the splitting fields in C for the polynomials (over Q).
a) x^3 - 1
b) x^4 - 1
c) x^3 + 3x^2 + 3x - 4
a) x^3 = 1
x = 1
also, x^3 - 1 = (x - 1)(x^2 + x + 1)
w = (-1 + √-3)/2 = -1/2 + (√3)/2i, where w is a root of x^2 + x + 1
Correct, but in fact
b) x^4 = 1
x = 1
also, x^4 - 1 = (x^2 - 1)(x^2 + 1)
This is incorrect: since both roots of are rational, we only need the roots of ,
which are , so here the splitting field is
c) I am not sure about.
Using basic calculus it's easy to see that this pol. has a real non-rational root between
0 and 1 and, since it is a monotone ascending function of x, that one is the only real root, so here
the splitting field is , with r the real root and w one of the two conjugate
complex non-real roots.
Can anyone let me know if my answer for a & b are correct and if not give me some help. I also need help with c.
Thanks in advance.