Splitting Fields

• March 14th 2011, 08:35 AM
page929
Splitting Fields
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)

b) x^4 = 1
x = 1
also, x^4 - 1 = (x^2 - 1)(x^2 + 1)
Q(1,√i)

c) I am not sure about.

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.

• March 14th 2011, 11:40 AM
tonio
Quote:

Originally Posted by 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 $\mathbb{Q}(1,w)=\mathbb{Q}(w)\,,\,w^3=1\,,\,w\neq 1$

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

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

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