Re: [math-fun] conjugation of algebraic numbers
Rich writes: << A curiosity: sqrt 6 + sqrt 10 + sqrt 15 is NOT a conjugate of - sqrt 6 - sqrt 10 - sqrt 15. The minimal polynomial of the first expression is X^4 - 62 X^2 - 240 X - 239. (Yet another property of 239.) On the other hand, sqrt 2 + sqrt 3 + sqrt 5 IS a conjugate of - sqrt 2 - sqrt 3 - sqrt 5. and the minimal polynomial is degree 8.
Hmmm, so is this curiosity always true for +-(sqrt(pq + sqrt(qr) + sqrt(pr)) and +-(sqrt(p) + sqrt(q) + sqrt(r)) . . . when p,q,r are distinct primes? --Dan P.S. Nice job of reverse engineering, Michael!
--- dasimov@earthlink.net wrote:
Rich writes:
<< A curiosity:
sqrt 6 + sqrt 10 + sqrt 15 is NOT a conjugate of - sqrt 6 - sqrt 10 - sqrt 15.
The minimal polynomial of the first expression is X^4 - 62 X^2 - 240 X - 239. (Yet another property of 239.)
On the other hand,
sqrt 2 + sqrt 3 + sqrt 5 IS a conjugate of - sqrt 2 - sqrt 3 - sqrt 5.
and the minimal polynomial is degree 8.
Hmmm, so is this curiosity always true for +-(sqrt(pq + sqrt(qr) + sqrt(pr)) and +-(sqrt(p) + sqrt(q) + sqrt(r)) . . . when p,q,r are distinct primes?
--Dan
Expand as a polynomial in x: (x-(u+v+w))*(x-(u-v-w))*(x-(-u+v-w))*(x-(-u-v+w)). The critical term is -(8uvw)x. The minimal polynomial for u+v+w has degree 4 when uvw is rational. So this explains Dan's observation. Gene __________________________________________________ Do You Yahoo!? Tired of spam? Yahoo! Mail has the best spam protection around http://mail.yahoo.com
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dasimov＠earthlink.net -
Eugene Salamin