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how to run CyclotomicField([zeta7+zeta7^-1]?

asked 2013-12-06 19:34:12 -0500

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 - LL.<x>=CyclotomicField(7);LL
   Cyclotomic Field of order 7 and
   degree 6

   x.conjugate()
   -x^5 - x^4 - x^3 - x^2 - x - 1

   LL.<c>=CyclotomicField(-x^5 - x^4 - x^3 - x^2  - 1);LL
   Traceback (click to the left of this block for traceback)
   ...
   TypeError: Unable to
   coerce -x^5 - x^4 - x^3 - x^2 - 1 to
   an integer

   s=LL.gen() + (LL.gen()).conjugate();s
   LL.<y>=CyclotomicField(s);LL
   Traceback (click to the left of this
   block for traceback) ... TypeError:
   Unable to coerce -x^5 - x^4 - x^3 -
   x^2 - 1 to an integer

   KK.<s> = NumberField(-x^5 - x^4 - x^3 - x^2 - 1); KK
   Traceback (click to the left of this block for traceback)
   ... NotImplementedError: number
   fields for non-monic polynomials not
   yet implemented.



   KKK.<s> = NumberField(x^5 + x^4 + x^3 + x^2 + 1); KKK
   Number Field in s with defining polynomial x^5 + x^4 +
   x^3 + x^2 + 1

KKK is same with Q[zeta7+zeta7^-1]?

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Comments

@cjsh: Please format your posts properly next time. Most likely you will get a reply sooner (if someone who knows this stuff happens to read it).

ppurka gravatar imageppurka ( 2013-12-06 23:27:33 -0500 )edit

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answered 2013-12-07 00:24:10 -0500

tmonteil gravatar image

First, the way you define Cyclotomic Fields leads to errors, since the parameter should be an integer, not a polynomial. You can read the documentation about Cyclotomic Fields by typing:

sage: CyclotomicField?

Then, here is how you can check that NumberField(x^5 + x^4 + x^3 + x^2 + 1) and $\mathbb{Q}[\zeta_7+\zeta_7^{-1}]$ are not equal.

sage: LL = CyclotomicField(7) ; LL
Cyclotomic Field of order 7 and degree 6

Note that the generator zeta7 was injected into the namespace:

sage: zeta7^6
-zeta7^5 - zeta7^4 - zeta7^3 - zeta7^2 - zeta7 - 1
sage: zeta7^7
1

sage: KKK.<s> = NumberField(x^5 + x^4 + x^3 + x^2 + 1); KKK
Number Field in s with defining polynomial x^5 + x^4 + x^3 + x^2 + 1

sage: MM = QQ[zeta7+zeta7^-1] ; MM
Number Field in a with defining polynomial x^3 + x^2 - 2*x - 1

You can check that MM and KKK are not equal since they do not have the same dimension over $\mathbb{Q}$ (which corresponds to the degree of the irreducible polynomial defining them):

sage: KKK.degree()
5
sage: MM.degree()
3

As a WARNING, you should notice that:

sage: KKK == MM
False

is not safe, since the equality also considers the name of the variable:

sage: KK.<t> = NumberField(x^5 + x^4 + x^3 + x^2 + 1); KK
Number Field in t with defining polynomial x^5 + x^4 + x^3 + x^2 + 1
sage: KKK.<s> = NumberField(x^5 + x^4 + x^3 + x^2 + 1); KKK
Number Field in s with defining polynomial x^5 + x^4 + x^3 + x^2 + 1
sage: KK == KKK
False
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thank you very much! MK.<zeta> = NumberField(x^3 + x^2 - 2*x - 1);MK MK.is_isomorphic(MM) (True,)

cjsh gravatar imagecjsh ( 2013-12-07 18:40:00 -0500 )edit

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Asked: 2013-12-06 19:34:12 -0500

Seen: 74 times

Last updated: Dec 07 '13