Abstractly constructing splitting fields I have a series of exercises where I have to determine the degree of various splitting fields. I am freely using the following observation, which I feel is intuitively true, but I am asking here to make sure it's true.


  Let $f(x) \in \mathbb Q[x]$. Let $a_1,\dots,a_n$ be roots of $f$ in $\mathbb C$. Then the splitting field is $\mathbb Q(a_1,\dots,a_n)$. 


Clearly $f$ splits over this extension, but I'm not sure how to "rigorously" check that this is indeed the smallest extension.
It's by definition. Recheck your definition of what $\mathbb{Q}(a)$ means! It's already the smallest such field.
The proposed minimal spliff (splitting field) is indeed minimal because it by definition is the minimal extension of the rationals which contains the roots.