template<typename NT, typename ROOT, typename DifferentExtensionComparable = Tag_false, typename FilterPredicates = Tag_false>
class CGAL::Sqrt_extension< NT, ROOT, DifferentExtensionComparable, FilterPredicates >

An instance of this class represents an extension of the type NT by one square root of the type ROOT.

NT is required to be constructible from ROOT.

NT is required to be an IntegralDomainWithoutDivision.

Sqrt_extension is RealEmbeddable if NT is RealEmbeddable.

For example, let Integer be some type representing $\Z$, then Sqrt_extension<Integer,Integer> is able to represent $\Z[\sqrt{\mathrm{root}}]$ for some arbitrary Integer $\mathrm{root}$. ^[1] The value of $\mathrm{root}$ is set at construction time, or set to zero if it is not specified.

Arithmetic operations among different extensions, say $\Z[\sqrt{a}]$ and $\Z[\sqrt{b}]$, are not supported. The result would be in $\Z[\sqrt{a},\sqrt{b}]$, which is not representable by Sqrt_extension<Integer,Integer>.

Attention

The user is responsible to check that arithmetic operations are carried out for elements from the same extensions only.

This is not tested by Sqrt_extension for efficiency reasons. A violation of the precondition leads to undefined behavior. Be aware that for efficiency reasons the given $\mathrm{root}$ is stored as it is given to the constructor. In particular, an extension by a square root of a square is considered as an extension.

Since elements of Sqrt_extension that lie in different extensions are not interoperable with respect to any arithmetic operations, the full value range of Sqrt_extension does not represent an algebraic structure. However, each subset of the value range that represents the extension of NT by a particular square root is a valid algebraic structure, since this subset is closed under all provided arithmetic operations. From there, Sqrt_extension can be used as if it were a model of an algebraic structure concept, with the following correspondence:

The extension of a UniqueFactorizationDomain or EuclideanRing is just an IntegralDomain, since the extension in general destroys the unique factorization property. For instance consider $\Z[\sqrt{10}]$, the extension of $\Z$ by $\sqrt{10}$: in $\Z[\sqrt{10}]$ the element 10 has two different factorizations $\sqrt{10} \cdot \sqrt{10}$ and $2 \cdot 5$. In particular, the factorization is not unique.

If NT is a model of RealEmbeddable the type Sqrt_extension is also considered as RealEmbeddable. However, by default it is not allowed to compare values from different extensions for efficiency reasons. In case such a comparison becomes necessary, use the member function compare with the according Boolean flag. If such a comparison is a very frequent case, override the default of DifferentExtensionComparable by giving CGAL::Tag_true as third template parameter. This effects the behavior of compare functions as well as the compare operators.

The fourth template argument, FilterPredicates, triggers an internal filter that may speed up comparisons and sign computations. In case FilterPredicates is set to CGAL::Tag_true the type first computes a double interval containing the represented number and tries to perform the comparison or sign computation using this interval. Once computed, this interval is stored by the corresponding Sqrt_extension object for further usage. Note that this internal filter is switched off by default, since it may conflict with other filtering methods, such as Lazy_exact_nt<Sqrt_extension>.

In case NT is not RealEmbeddable, DifferentExtensionComparable as well as FilterPredicates have no effect.

Is Model Of:

Assignable

CopyConstructible

DefaultConstructible

EqualityComparable

ImplicitInteroperable with int

ImplicitInteroperable with NT

Fraction if NT is a Fraction

RootOf_2

See Also

IntegralDomainWithoutDivision

IntegralDomain

Field

RealEmbeddable

CGAL::Tag_true

CGAL::Tag_false