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Units in number fields.

asked 2015-02-01 11:13:14 -0500

juan gravatar image

updated 2015-02-01 11:17:25 -0500

tmonteil gravatar image

I want to get the expression of a unit in a number field in terms of the base by means of U.log(). Here is my code:

K.<z> = CyclotomicField(21)
U = UnitGroup(K)
U.fundamental_units()
a=(z^3+z^6+z^12)/2+1      # a is  (3 + i sqrt[7])/4   so it is a unit in K
U.log(a)

I get the error

ValueError: 1/2*z^11 - 1/2*z^9 + 1/2*z^8 + 1/2*z^4 + 1/2*z + 1/2 is not a unit

But it is a unit. It is true that a = 1/2*z^11 - 1/2*z^9 + 1/2*z^8 + 1/2*z^4 + 1/2*z + 1/2.

What I'm doing wrong ?

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@tmonteil Thanks for the edits.

juan gravatar imagejuan ( 2015-02-01 11:22:48 -0500 )edit

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answered 2015-02-01 11:57:36 -0500

Francis Clarke gravatar image

The problem is that your ais not an algebraic integer (though its norm is 1).
Following on from your code :

sage: a.is_integral()
False
sage: a.minpoly()
x^2 - 3/2*x + 1
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Asked: 2015-02-01 11:13:14 -0500

Seen: 51 times

Last updated: Feb 01 '15