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(1-i)^(1/3),(1-i)^(1/4) is algebra interger?

asked 2013-12-06 19:56:42 -0600

cjsh gravatar image

updated 2015-07-31 11:16:30 -0600

FrédéricC gravatar image

I try http://www.wolframalpha.com/input/?i=... answer:UNKNOW

but sagemath :true,why?which is right?

K.<a> = NumberField((x^3-1)^2+1);K;a.list();a.is_integral() 

Number Field in a with defining polynomial x^6 - 2*x^3 + 2
[0, 1, 0, 0, 0, 0]
True
K.<a> = NumberField((x^4-1)^2+1);K;a.list();a.is_integral() 

Number Field in a with defining polynomial x^8 - 2*x^4 + 2
[0, 1, 0, 0, 0, 0, 0, 0]
True
K.<a> = NumberField((x^5-1)^2+1);K;a.list();a.is_integral() 

Number Field in a with defining polynomial x^10 - 2*x^5 + 2
[0, 1, 0, 0, 0, 0, 0, 0, 0, 0]
True
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answered 2013-12-06 23:25:59 -0600

tmonteil gravatar image

updated 2013-12-06 23:52:26 -0600

In your case, you can get conviced by typing:

sage: P = a.absolute_minpoly() ; P
x^6 - 2*x^3 + 2
sage: P(a)
0
sage: P in ZZ[x]
True
sage: P.is_monic()
True

https://en.wikipedia.org/wiki/Algebra...

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thank you very much!

cjsh gravatar imagecjsh ( 2013-12-07 18:41:15 -0600 )edit

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Asked: 2013-12-06 19:56:42 -0600

Seen: 86 times

Last updated: Dec 07 '13