I have to find the degree of the field extension $\mathbb Q(2^{1/2}, 2^{1/4}, 2^{1/8})$ over $\mathbb Q$.
Hope to get a elaborate answer. (I am very much new to field extension, so I would like to see the steps and understand them, which will help me in future.) Thanks.
$\mathbb{Q}(2^{1/8})$ already contains $2^{1/4}$ and $2^{1/2}$ so you do not need to include those. The minimal polynomial for $t=2^{1/8}$ in $\mathbb{Q}$ is $t^8-2=0$. The Eisenstein test confirms that it's irreducible. Hence, elements $\{1,t, t^2, \ldots, t^7\}$ are linearly independent and form a basis of your extension over $\mathbb{Q}$ which means its degree is $8$.