Finding the minimal polynomial over Q

  • #26
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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?
 
  • #27
784
11
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.

Also, how do I know which numbers are constructible in q(r) where r is the third primitive root of unit multiplied by the fourth real root of 2?
 
  • #28
20
5
PsychonautQQ

You may have already figured it out, but;
Don't have to cube expression after expanding c4; & it happens that product of degrees of elements just give the max possible degree for poly, it can happen to be less than that when roots span common subspace. Once you have only elements from base field in expression your done, don't necessarily need to try for max degree. So, look at, c4 = c3*c remember what c3 is, and see you just need to get rid of 'i' & square root in just one term.

As for your other post about constructibility, I really have to dig deep, been a long-time. Think generally if the extension generated by an element is of degree 2 or a power of 2, that it's constructible. Not sure if any special exceptions when roots of unity are involved, that are not of degree a power of 2; should get another opinion to be sure.
 
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