PysychonautQQ is a pretty cool gamer tag. Almost did not see this before I left work; no internet connection available where I live; not yet anyway.
But, towards your question. Yeah it does not always work so well, but it's good to check if method will work since it goes pretty quick.
Gonna try using formatting given here, not how to check if I did it right.
If I'm reading it right your trying to get min poly for r = c*24 where c = (-1 + i*31/2)/2 or (-1 - i*31/2)/2?
Then look at r4 = (c*21/4)4 = c4*2, expand c4 and separate terms in your field from those that are not; should work like last one from this point. Will illustrate example from the last problem: from (r^2 - 1) = 2i*6^(1/2) square both sides gives (r^2 -1)^2 = -24, then expand & simplify.
Have to sign off till tomorrow, hope this helps.
Thanks for all the help. I'm a pretty slow learner and I just want to clarify: After I expand c^4 and separate the terms, I then have to cube both sides so I have a polynomial with r^12, correct?