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I try http://www.wolframalpha.com/input/?i=Is+cbrt%281%2Bi%29+an+algebraic+integer%3F answer:UNKNOW

but sagemath :true,why?which is right?

K. = 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. = 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. = NumberField((x^5-1)^2+1);K;a.list();a.is_integral()

I try http://www.wolframalpha.com/input/?i=Is+cbrt%281%2Bi%29+an+algebraic+integer%3F 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. 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. 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]
TrueTrue

I try http://www.wolframalpha.com/input/?i=Is+cbrt%281%2Bi%29+an+algebraic+integer%3F 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

I try http://www.wolframalpha.com/input/?i=Is+cbrt%281%2Bi%29+an+algebraic+integer%3F 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