Definition of $\mathbb{Q}(\theta)$

Question

I have previously seen that $\mathbb{Q}(\theta) = ${$a+b\sqrt\theta:a,b \in \mathbb{Q}$}.

But I saw an example $\mathbb{Q}(\sqrt[3]{2})=${$a+b\sqrt[3]{2}+c\sqrt[3]{4} : a,b,c\in\mathbb{Q}$}.

I was wondering what is the general rule here for when we define these structures.

Answer 1

Let $\theta$ be an algebraic number and $f(x) \in \mathbb{Q}[x]$ its minimal polynomial. Suppose $\deg(f)=n$: then $1, \theta,...,\theta^{n-1}$ form a $\mathbb{Q}$-basis for $\mathbb{Q}(\theta)$.

In your case $x^3-2$ is the minimal polynomial of $\sqrt[3]{2}$ and a $\mathbb{Q}$-basis of $\mathbb{Q}(\sqrt[3]{2})$ is given by $1, \sqrt[3]{2},\sqrt[3]{2}^2=\sqrt[3]{4}$.