Question: Determine the degree of $\mathbb{Q}(\alpha)$ over $\mathbb{Q}$, where $\alpha^3=2$. Determine the degree of the splitting field of $f(t) = t^3 - 2$ over $\mathbb{Q}$.
Is there a difference between these two questions? To answer the first part, I attempted to say that $\alpha$ is algebraic over $\mathbb{Q}$ since it is a solution of $f(t)$, and since it is irreducible over $\mathbb{Q}$ of degree $3$, then the degree of $\mathbb{Q}(\alpha)$ over $\mathbb{Q}$ should be $3$. Can anyone use simple terms to explain how to answer these types of questions? I feel lost!