Trivial valuations - Discrete Valuations and Discrete Pseudo-Valuations

Skip to content
Back to top
View this page
Edit this page

# Trivial valuations

AUTHORS:

- Julian Rüth (2016-10-14): initial version

EXAMPLES:

Sage

    sage: v = valuations.TrivialValuation(QQ); v
    Trivial valuation on Rational Field
    sage: v(1)
    0

Python

    >>> from sage.all import *
    >>> v = valuations.TrivialValuation(QQ); v
    Trivial valuation on Rational Field
    >>> v(Integer(1))
    0

class sage.rings.valuation.trivial_valuation.TrivialDiscretePseudoValuation(parent)

  Bases: `TrivialDiscretePseudoValuation_base`, `InfiniteDiscretePseudoValuation`

  The trivial pseudo-valuation that is \(\infty\) everywhere.

  EXAMPLES:

  Sage

      sage: v = valuations.TrivialPseudoValuation(QQ); v
      Trivial pseudo-valuation on Rational Field

  Python

      >>> from sage.all import *
      >>> v = valuations.TrivialPseudoValuation(QQ); v
      Trivial pseudo-valuation on Rational Field

  lift(X)

    Return a lift of `X` to the domain of this valuation.

    EXAMPLES:

    Sage

        sage: v = valuations.TrivialPseudoValuation(QQ)
        sage: v.lift(v.residue_ring().zero())
        0

    Python

        >>> from sage.all import *
        >>> v = valuations.TrivialPseudoValuation(QQ)
        >>> v.lift(v.residue_ring().zero())
        0

  reduce(x)

    Reduce `x` modulo the positive elements of this valuation.

    EXAMPLES:

    Sage

        sage: v = valuations.TrivialPseudoValuation(QQ)
        sage: v.reduce(1)
        0

    Python

        >>> from sage.all import *
        >>> v = valuations.TrivialPseudoValuation(QQ)
        >>> v.reduce(Integer(1))
        0

  residue_ring()

    Return the residue ring of this valuation.

    EXAMPLES:

    Sage

        sage: valuations.TrivialPseudoValuation(QQ).residue_ring()
        Quotient of Rational Field by the ideal (1)

    Python

        >>> from sage.all import *
        >>> valuations.TrivialPseudoValuation(QQ).residue_ring()
        Quotient of Rational Field by the ideal (1)

  value_group()

    Return the value group of this valuation.

    EXAMPLES:

    A trivial discrete pseudo-valuation has no value group:

    Sage

        sage: v = valuations.TrivialPseudoValuation(QQ)
        sage: v.value_group()
        Traceback (most recent call last):
        ...
        ValueError: The trivial pseudo-valuation that is infinity everywhere does not have a value group.

    Python

        >>> from sage.all import *
        >>> v = valuations.TrivialPseudoValuation(QQ)
        >>> v.value_group()
        Traceback (most recent call last):
        ...
        ValueError: The trivial pseudo-valuation that is infinity everywhere does not have a value group.

class sage.rings.valuation.trivial_valuation.TrivialDiscretePseudoValuation_base(parent)

  Bases: `DiscretePseudoValuation`

  Base class for code shared by trivial valuations.

  EXAMPLES:

  Sage

      sage: v = valuations.TrivialPseudoValuation(ZZ); v
      Trivial pseudo-valuation on Integer Ring

  Python

      >>> from sage.all import *
      >>> v = valuations.TrivialPseudoValuation(ZZ); v
      Trivial pseudo-valuation on Integer Ring

  is_negative_pseudo_valuation()

    Return whether this valuation attains the value \(-\infty\).

    EXAMPLES:

    Sage

        sage: v = valuations.TrivialPseudoValuation(QQ)
        sage: v.is_negative_pseudo_valuation()
        False

    Python

        >>> from sage.all import *
        >>> v = valuations.TrivialPseudoValuation(QQ)
        >>> v.is_negative_pseudo_valuation()
        False

  is_trivial()

    Return whether this valuation is trivial.

    EXAMPLES:

    Sage

        sage: v = valuations.TrivialPseudoValuation(QQ)
        sage: v.is_trivial()
        True

    Python

        >>> from sage.all import *
        >>> v = valuations.TrivialPseudoValuation(QQ)
        >>> v.is_trivial()
        True

  uniformizer()

    Return a uniformizing element for this valuation.

    EXAMPLES:

    Sage

        sage: v = valuations.TrivialPseudoValuation(ZZ)
        sage: v.uniformizer()
        Traceback (most recent call last):
        ...
        ValueError: Trivial valuations do not define a uniformizing element

    Python

        >>> from sage.all import *
        >>> v = valuations.TrivialPseudoValuation(ZZ)
        >>> v.uniformizer()
        Traceback (most recent call last):
        ...
        ValueError: Trivial valuations do not define a uniformizing element

class sage.rings.valuation.trivial_valuation.TrivialDiscreteValuation(parent)

  Bases: `TrivialDiscretePseudoValuation_base`, `DiscreteValuation`

  The trivial valuation that is zero on nonzero elements.

  EXAMPLES:

  Sage

      sage: v = valuations.TrivialValuation(QQ); v
      Trivial valuation on Rational Field

  Python

      >>> from sage.all import *
      >>> v = valuations.TrivialValuation(QQ); v
      Trivial valuation on Rational Field

  extensions(ring)

    Return the unique extension of this valuation to `ring`.

    EXAMPLES:

    Sage

        sage: v = valuations.TrivialValuation(ZZ)
        sage: v.extensions(QQ)
        [Trivial valuation on Rational Field]

    Python

        >>> from sage.all import *
        >>> v = valuations.TrivialValuation(ZZ)
        >>> v.extensions(QQ)
        [Trivial valuation on Rational Field]

  lift(X)

    Return a lift of `X` to the domain of this valuation.

    EXAMPLES:

    Sage

        sage: v = valuations.TrivialValuation(QQ)
        sage: v.lift(v.residue_ring().zero())
        0

    Python

        >>> from sage.all import *
        >>> v = valuations.TrivialValuation(QQ)
        >>> v.lift(v.residue_ring().zero())
        0

  reduce(x)

    Reduce `x` modulo the positive elements of this valuation.

    EXAMPLES:

    Sage

        sage: v = valuations.TrivialValuation(QQ)
        sage: v.reduce(1)
        1

    Python

        >>> from sage.all import *
        >>> v = valuations.TrivialValuation(QQ)
        >>> v.reduce(Integer(1))
        1

  residue_ring()

    Return the residue ring of this valuation.

    EXAMPLES:

    Sage

        sage: valuations.TrivialValuation(QQ).residue_ring()
        Rational Field

    Python

        >>> from sage.all import *
        >>> valuations.TrivialValuation(QQ).residue_ring()
        Rational Field

  value_group()

    Return the value group of this valuation.

    EXAMPLES:

    A trivial discrete valuation has a trivial value group:

    Sage

        sage: v = valuations.TrivialValuation(QQ)
        sage: v.value_group()
        Trivial Additive Abelian Group

    Python

        >>> from sage.all import *
        >>> v = valuations.TrivialValuation(QQ)
        >>> v.value_group()
        Trivial Additive Abelian Group

class sage.rings.valuation.trivial_valuation.TrivialValuationFactory(clazz, parent, *args, **kwargs)

  Bases: `UniqueFactory`

  Create a trivial valuation on `domain`.

  EXAMPLES:

  Sage

      sage: v = valuations.TrivialValuation(QQ); v
      Trivial valuation on Rational Field
      sage: v(1)
      0

  Python

      >>> from sage.all import *
      >>> v = valuations.TrivialValuation(QQ); v
      Trivial valuation on Rational Field
      >>> v(Integer(1))
      0

  create_key(domain)

    Create a key that identifies this valuation.

    EXAMPLES:

    Sage

        sage: valuations.TrivialValuation(QQ) is valuations.TrivialValuation(QQ) # indirect doctest
        True

    Python

        >>> from sage.all import *
        >>> valuations.TrivialValuation(QQ) is valuations.TrivialValuation(QQ) # indirect doctest
        True

  create_object(version, key, **extra_args)

    Create a trivial valuation from `key`.

    EXAMPLES:

    Sage

        sage: valuations.TrivialValuation(QQ) # indirect doctest
        Trivial valuation on Rational Field

    Python

        >>> from sage.all import *
        >>> valuations.TrivialValuation(QQ) # indirect doctest
        Trivial valuation on Rational Field