I was wondering how we find exactly which numbers are constructible numbers in a field $ℚ(a)$ where $ a = ζ \sqrt{2}$ and ζ is an nth root of unity.
For example the primitive $3$-rd root of unity...
If I know that the degree of the extension $[ℚ(ζ, \sqrt{2}): ℚ] = 4$ can I conclude that the constructible elements would be those numbers of $ℚ(a)$ which have a degree which is a power of 2? Is it enough to just conclude that or can we find the actual numbers?
Also I was wondering if this extends for all constructible $ a = ζ \sqrt[b]{2}$ i.e. $ a = ζ \sqrt[3]{2}$?
Any comments or discussions would be great thank you.