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moved motivation for Field refining IntegralDomain and not EuclideanRing to 

users manual
This commit is contained in:
Michael Hemmer 2007-01-29 08:51:32 +00:00
parent d97ace70d0
commit ba37f9906f
2 changed files with 22 additions and 9 deletions
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7
Algebraic_foundations/doc_tex/Algebraic_foundations/algebraic_structures.tex
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@ -41,6 +41,13 @@ corresponds to integral domains in the algebraic sense, the
distinction results from the fact that some implementations of
integral domains lack the (algebraically always well defined) integral
division.
Note that \ccc{Field} refines \ccc{IntegralDomain}. This is because
most ring-theoretic notions like greatest common divisors become trivial for
\ccc{Field}s. Hence we see \ccc{Field} as a refinement of
\ccc{IntegralDomain} and not as a
refinement of one of the more advanced types of ring.
If an algorithm wants to rely on gcd or remainder computation, it is trying
to do things it shouldn't do with a \ccc{Field} in the first place.
The main properties of an algebraic structure are collected in the class

24
Algebraic_foundations/doc_tex/Algebraic_foundations_ref/Field.tex
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@ -3,18 +3,24 @@
\ccDefinition
A model of \ccc{Field} is an IntegralDomain in which every non-zero element has a multiplicative inverse.
Thus, one can divide by any non-zero element. Hence division is defined for any divisor != 0.
For a Field, we require this division operation to be available through operators / and /=.
A model of \ccc{Field} is an\ccc{IntegralDomain} in which every non-zero element
has a multiplicative inverse.
Thus, one can divide by any non-zero element.
Hence division is defined for any divisor != 0.
For a Field, we require this division operation to be available through
operators / and /=.
Moreover, \ccc{CGAL::Algebraic_structure_traits< Field >} is a model of \ccc{AlgebraicStructureTraits} providing:\\
- \ccc{CGAL::Algebraic_structure_traits< Field >::Algebraic_type} derived from \ccc{Field_tag} \\
Moreover, \ccc{CGAL::Algebraic_structure_traits< Field >} is a model of
\ccc{AlgebraicStructureTraits} providing:\\
- \ccc{CGAL::Algebraic_structure_traits< Field >::Algebraic_type} derived
from \ccc{Field_tag} \\
\ccHeading{Remarks:}
Most ring-theoretic notions like greatest common divisors become trivial for fields.
Hence we see Field as a refinement of IntegralDomain and not as a refinement of one of the more advanced
types of ring. If an algorithm wants to rely on gcd or remainder computation, it is trying to do things
it shouldn't do with a field in the first place.
Most ring-theoretic notions like greatest common divisors become trivial for
fields. Hence we see Field as a refinement of IntegralDomain and not as a
refinement of one of the more advanced types of ring.
If an algorithm wants to rely on gcd or remainder computation, it is trying
to do things it shouldn't do with a field in the first place.
\ccRefines
\ccc{IntegralDomain}