I'm sorry if I sound too ignorant. I don't have a high level of knowledge in math.
While reading an article about Galois theory, I've become confused over the concept of a splitting field. The author defines the splitting Field of the polynomial $p(x)$ as the smallest field extension of $Q$ that contains all the roots of $p(x)$. He/She then gives the examples of $p(x)=x^2−2$, which splitting Field is $\mathbb{Q}[\sqrt2]$ since "it contains all the roots of $p(x)$ and if it had fewer elements it either wouldn't contain all the roots or wouldn't be a field" and that of $p(x)=x^4−5x^2+6$ which splitting field is $\mathbb{Q}[\sqrt 2,\sqrt 3]$, then the author decides to finish the paragraph by asking the question "Can you see why?".
And that is the issue, I cannot see why :/
I struggled for a while trying to think of a general method or explanation to why those field extensions correspond to those polynomials, but I couldn't come up with anything.
Any thoughts/ideas would be really appreciated!