# Galois group

• May 6th 2010, 07:27 AM
KSM08
Galois group
Does anyone know how to find the Galois group of x^3-3 over the finite field F_13?
• May 6th 2010, 07:53 AM
tonio
Quote:

Originally Posted by KSM08
Does anyone know how to find the Galois group of x^3-3 over the finite field F_13?

As $\displaystyle \sqrt[3]{3}\notin \mathbb{F}_{13}\,\,\,and\,\,\,3^3=9^3=1\!\!\!\pmod {13}$ , the roots of $\displaystyle f(x)=x^3-3$ are $\displaystyle w:=\sqrt[3]{3}\,,\,3w\,,\,9w$ , and from here that the spliiting field of

$\displaystyle f(x)$ over $\displaystyle \mathbb{F}_{13}$ is $\displaystyle K:=\mathbb{F}_{13}(w)\Longrightarrow [K:\mathbb{F}_{13}]=3$ ...

Tonio
• May 7th 2010, 12:42 AM
KSM08
Thanks- I had worked out that the degree of the splitting field is three, but what does this tell me? Since the polynomial is not separable, presumably it's not the case that the size of the Galois group is 3. (But that 3 divides the size?) What automorphisms should I be trying?
• May 7th 2010, 01:01 AM
kompik
Theorem 9.5.1 in Roman's Field Theory
I think Theorem 9.5.1 here answers your question.
• May 7th 2010, 01:03 AM
NonCommAlg
since $\displaystyle x^3-3$ is irreducible over $\displaystyle \mathbb{F}_{13},$ the splitting field of this polynomial is $\displaystyle \frac{\mathbb{F}_{13}[x]}{(x^3-3)} \cong \mathbb{F}_{13^3}=K.$ thus $\displaystyle 3=[K:\mathbb{F}_{13}]=|Gal(K/\mathbb{F}_{13})|$ and so the Galois group is just $\displaystyle \mathbb{Z}/3\mathbb{Z}.$
• May 7th 2010, 03:27 AM
tonio
Quote:

Originally Posted by KSM08
Thanks- I had worked out that the degree of the splitting field is three, but what does this tell me? Since the polynomial is not separable,

Why do you think so?? Of course it is separable, and if you have any doubt just remember the basic theorem: any finite extension of a finite field (or any extension of a characteristic zero field) is separable.
Furthermore, if you already knew the degree of the splitting field then, this being a prime, doesn't leave you many choices for the Galois group, does it? (Giggle)

Tonio

presumably it's not the case that the size of the Galois group is 3. (But that 3 divides the size?) What automorphisms should I be trying?

.