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**page929** 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 $\displaystyle \mathbb{Q}(1,w)=\mathbb{Q}(w)\,,\,w^3=1\,,\,w\neq 1$

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 $\displaystyle x^2-1$ are rational, we only need the roots of $\displaystyle x^2+1$ ,

which are $\displaystyle \pm i=\pm \sqrt{-1}$ , so here the splitting field is $\displaystyle \mathbb{Q}(i)$

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 $\displaystyle \mathbb{Q}(r,w)$ , 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.