# Splitting field for X^3-5

• May 18th 2013, 03:52 AM
Ant
Splitting field for X^3-5
Hi,

Let $\displaystyle f=X^3 -5$, the $\displaystyle \Sigma_f = \mathbb{Q}[\alpha,\omega]$ with $\displaystyle \alpha$ real cube root of 5 and $\displaystyle \omega$ primitive root of 1.

$\displaystyle [\mathbb{Q}(\alpha):\mathbb{Q}]=3$ because $\displaystyle X^3-5$ is irreducible.

But what is $\displaystyle [\mathbb{Q}(\alpha,\omega):\mathbb{Q}(\alpha)]$?

$\displaystyle [\mathbb{Q}(\alpha,\omega):\mathbb{Q}(\alpha)] >1$ because $\displaystyle \omega \in \mathbb{C} \setminus \mathbb{R}$ and yet $\displaystyle \mathbb{Q}(\alpha)$ is a subfield of $\displaystyle \mathbb{R}$. Also the degree is $\displaystyle \leq 3$ because $\displaystyle X^3 -1$ has $\displaystyle \omega$ as a root. So the possibilities are 2 or 3.

If it's 2. Then because the min poly has to be monic and $\displaystyle \omega^2 + \omega = \sqrt{2}$

I think that we have to express $\displaystyle \sqrt{2}$ as a linear combination (with rational coefficients) of $\displaystyle \alpha$ and $\displaystyle \alpha^2$.

I can't see how do to this?

Thanks for any help!
• May 18th 2013, 08:46 AM
Siron
Re: Splitting field for X^3-5
The roots of $\displaystyle X^3-1$ are $\displaystyle X_1=1, X_2=e^{\frac{2 \pi i}{3}}, X_3 = e^{\frac{4 \pi i}{3}}$ whereby $\displaystyle X_2$ and $\displaystyle X_3$ have minimal polynomial $\displaystyle T^2+T+1$
• May 18th 2013, 11:01 AM
Ant
Re: Splitting field for X^3-5
Of course, I made a stupid mistake:

$\displaystyle \omega^2 + \omega = -1$!

thanks