Degree of field extensions $\mathbb Q(\alpha)/\mathbb Q$ and $\mathbb Q(\alpha^{12})/\mathbb Q$, where $\alpha = 2^{1/3}+3^{1/4}$

Question

How do I find the degree of the following field extensions $\mathbb Q(\alpha)/\mathbb Q$ and $\mathbb Q(\alpha^{12})/\mathbb Q$, where $\alpha = 2^{1/3}+3^{1/4}$ ?

Note : Since $\mathbb Q(\alpha) \subseteq \mathbb Q(2^{1/3},3^{1/4})$ and $[\mathbb Q(2^{1/3},3^{1/4}):\mathbb Q]=12$, so $[\mathbb Q(\alpha):\mathbb Q]|12$ ; but then I have no clue how to proceed further.

Please help.