[SOLVED BUT CANNOT DELETE THREAD] finding the Galois Group G(Q(2,3)/Q)\mathbb{Q(\sqrt{2}, \sqrt3)}/\mathbb{Q})

[MathNugget]

I am trying to find the Galois Group G($\mathbb{Q(\sqrt{2}, \sqrt3)}/\mathbb{Q})$. That means, I have to find the automorphisms $\phi: \mathbb{Q(\sqrt{2}, \sqrt3)} \rightarrow \mathbb{Q(\sqrt{2}, \sqrt3)}$, for which $\phi(x)=x, \forall x \in \mathbb{Q}$.
Let's say K=$\mathbb{Q(\sqrt{2}, \sqrt3)}$.

I am fairly certain that all elements of K can be written as $a + b\sqrt2 +c\sqrt3 + d\sqrt 6$, $a, b, c, d \in \mathbb{Q}$.
I know that if $K=\mathbb{Q}(\sqrt{c})$, there are 2 automorphisms: $a+b\sqrt{c} \rightarrow a-b\sqrt{c}$ and identity. (let's say here that c is positive, and isn't the square of an integer or rational, so that $\sqrt{c}$ is irrational but not complex.
I also know that if $K=\mathbb{R}$ has just 1 automorphism satisfying this, the identity.

I have read the wikipedia page on Galois groups, there is a bit of info about it, but it doesn't satisfy my question. Would this field have 2^3 automorphisms? $a \pm b\sqrt2 \pm c\sqrt3 \pm d\sqrt6$?

 

[MathNugget]

Nevermind, found it...https://math.stackexchange.com/questions/455067/computing-galois-group-of-mathbbq-sqrt2-sqrt3-mathbbq
How can I delete thread?

 

[khansaheb]

Nevermind, found it...https://math.stackexchange.com/questions/455067/computing-galois-group-of-mathbbq-sqrt2-sqrt3-mathbbq
How can I delete thread?

You should not delete the thread - your reference URL can help the next student.

 

[MathNugget]

You should not delete the thread - your reference URL can help the next student.

alright. Do forum posts appear on google searches? I am not sure they do...

 

[Steven G]

Yes, they do appear on google searches.