I am following example 50.9 of Fraleigh's first course in abstract algebra, 7 edition.
I want to find the degree of the splitting field of $f(x) = x^3 - 2$ over $\mathbb{Q}$. I verified that $f(x)$ factors into a product of $(x-2^{1/3})(x^2 + 2^{1/3}x + 2^{2/3} )$. The author says (without doing any calculations) that $x^2 + 2^{1/3}x + 2^{2/3}$ is irreducible over $\mathbb{Q}(2^{1/3})$(how can he see this without doing a calculation?), and hence the splitting field of $x^2 + 2^{1/3}x + 2^{2/3}$ has degree 2 over $\mathbb{Q}(2^{1/3})$. How can he see this? Is the splitting field of every irreducible quadratic polynomial degree 2? Does this extend to polynomials of degree $n$ somehow?