Norm in UniversalCyclotomicField

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Thank you for your report, this is indeed a bug. Now tracked at

• Sage Trac ticket 26872: Implement abs for UniversalCyclotomicField elements

As a workaround,

• one could convert a to a vector over the field of algebraic numbers; then the norm is given as an exact algebraic number:

  sage: a = vector([E(8), E(7)])
  sage: b = a.change_ring(QQbar)
  sage: bn = b.norm()
  sage: bn
  1.414213562373095?
  sage: bn.parent()
  Algebraic Real Field

• one could convert a to a vector over a cyclotomic field (not the universal one); sadly the norm is then given as an inexact (floating-point) number:

  sage: a = vector([E(8), E(7)])
  sage: c = a.apply_map(lambda x: x.to_cyclotomic_field())
  sage: cn = c.norm()
  sage: cn
  1.41421356237310
  sage: cn.parent()
  Real Field with 53 bits of precision

• the square of the norm of the vector a can be obtained directly as a * a.conjugate(), but one then gets it as an element in the universal cyclotomic field, and a similar bug prevents from taking the square root of that directly:

  sage: a = vector([E(8), E(7)])
  sage: ann = a * a.conjugate()
  sage: ann
  2
  sage: ann.parent()
  Universal Cyclotomic Field
  sage: an = sqrt(ann)
  Traceback (most recent call last)
  ...
  NotImplementedError: is_square() not implemented for elements of Universal Cyclotomic Field

  and one again has to go through a cyclotomic field other than the universal one, or the field of real algebraic numbers:

  sage: nna = AA(a*a.conjugate())
  sage: nna
  2
  sage: nna.parent()
  Algebraic Real Field
  sage: na = sqrt(nna)
  sage: na
  1.414213562373095?
  sage: na.parent()
  Algebraic Real Field