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!
You're right about the first part: the degree of $\mathbb Q(\alpha)$ over $\mathbb Q$ does equal 3. (For another explanation: every element in $\mathbb Q(\alpha)$ can be written uniquely in the form $x+y\alpha+z\alpha^2$ with $x,y,z\in\mathbb Q$, since $\alpha^3=2$; therefore $\mathbb Q(\alpha)$ is a $3$-dimensional vector space over $\mathbb Q$, as suspected.)
As for the second part: what is the splitting field of $f(t)$? I suspect that once you identify that mathematical object, you'll be able to see that the two questions are indeed different. What is the general definition of "splitting field"?