mirror of https://github.com/CGAL/cgal
39 lines
1.4 KiB
TeX
39 lines
1.4 KiB
TeX
\begin{ccRefConcept}{Field}
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\ccDefinition
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A model of \ccc{Field} is an IntegralDomain in which every non-zero element has a multiplicative inverse.
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Thus, one can divide by any non-zero element. Hence division is defined for any divisor != 0.
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For a Field, we require this division operation to be available through operators / and /=.
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Moreover, \ccc{CGAL::Algebraic_structure_traits< Field >} is a model of \ccc{AlgebraicStructureTraits} providing:\\
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- \ccc{CGAL::Algebraic_structure_traits< Field >::Algebraic_type} derived from \ccc{Field_tag} \\
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\ccHeading{Remarks:}
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Most ring-theoretic notions like greatest common divisors become trivial for fields.
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Hence we see Field as a refinement of IntegralDomain and not as a refinement of one of the more advanced
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types of ring. If an algorithm wants to rely on gcd or remainder computation, it is trying to do things
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it shouldn't do with a field in the first place.
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\ccRefines
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\ccc{IntegralDomain}
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\ccOperations
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\ccFunction{Field operator/(const Field &a, const Field &b);}{}
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\ccGlue
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\ccMethod{Field operator/=(const Field &a);}{}
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\ccSeeAlso
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\ccRefIdfierPage{IntegralDomainWithoutDiv}\\
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\ccRefIdfierPage{IntegralDomain}\\
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\ccRefIdfierPage{UFDomain}\\
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\ccRefIdfierPage{EuclideanRing}\\
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\ccRefIdfierPage{Field}\\
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\ccRefIdfierPage{FieldWithSqrt}\\
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\ccRefIdfierPage{AlgebraicStructureTraits}\\
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\ccHasModels
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\end{ccRefConcept} |