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 **Q**(1,w) Correct, but in fact b) x^4 = 1 x = 1 also, x^4 - 1 = (x^2 - 1)(x^2 + 1) **Q**(1,√i) 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.

Tonio 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.