1Find all the conjugates of 213 w over Q where w is a comp

1Find all the conjugates of = 2^(1/3) + w over Q, where w is a complex root of 1; that is, w = 1/2 + ((3)/2))i.

2Find all the conjugates of over Q( 2^(1/3))

3Find all the conjugates of over Q(w).

[It may not be easy to directly find the minimal polynomial of . But you can find a relevant Galois group and apply its elements to .]

Solution

= 2^(1/3) + w where w = 1/2 + ((3)/2))i.
Thus = 2^(1/3) 1/2 + ((3)/2))i. = [2^1/3-1/2] + i [ 3/2]
Its conjugate will be  [2^1/3-1/2] - i [ 3/2]