Finding the Galois group of $f(t)=t^{3}-5t^{2}+11t-15$ over $\mathbb{Q}$.

Question

Ok, so I recently asked a very similar question here Finding the Galois group of $f(t)=t^{3}-4t+2$ over $\mathbb{Q}$.

However, $f(t)=t^{3}-5t^{2}+11t-15=(t-3)(t^{2}-2t+5)$. So the roots of $f$ are $3$, $1-2i$, and $1+2i$. Presumably I should consider the extension $\mathbb{Q}(3,1+2i,1-2i)$.

First of all, is this correct?

Second, am I right in saying that $\mathbb{Q}(3,1+2i,1-2i)=\mathbb{Q}(3,1+2i)$?

Third, if this is correct, where do I go from here?

Thank you.

Answer 1

You're right in saying that $\mathbb{Q}(3,1+2i,1-2i)=\mathbb{Q}(3,1+2i)$.

Now note that $\mathbb{Q}(3,1+2i)=\mathbb{Q}(i)$.

So the splitting field of $f$ is $\mathbb{Q}(i)$ and you have to find $\text{Gal}(\mathbb{Q}(i)/\mathbb{Q})$, which has order $2$...