dedekind::morphologies::Modular< N > Struct Template Reference

\(\mathbb{Z}/N\mathbb{Z}\) as a total finite cyclic ring. More...

Collaboration diagram for dedekind::morphologies::Modular< N >:

Public Types
using	machine_type = decltype(N)
using	Domain = Modular
using	wide_type

Public Member Functions
constexpr	Modular (machine_type v=0)
constexpr Modular	inverse () const
	The multiplicative inverse on a prime modulus, \(a^{N-2}\) (Fermat): ℤ/p is the field 𝔽ₚ.

Static Public Member Functions
static constexpr Modular	generator ()
static constexpr Modular	successor (Modular a)

Public Attributes
machine_type	value

Static Public Attributes
static constexpr machine_type	modulus = N
	The modulus N, as a value.

Friends
constexpr friend Modular	operator+ (Modular a, Modular b)
constexpr friend Modular	operator- (Modular a, Modular b)
constexpr friend Modular	operator* (Modular a, Modular b)
constexpr friend Modular	operator/ (Modular a, Modular b)
	Division on a prime modulus: multiplication by the inverse.
constexpr friend bool	operator== (Modular a, Modular b)
constexpr friend std::weak_ordering	constant_order (Modular a, Modular b)
	The structural order of residues as constants of a term (#884, canonical's sort key): their stored representatives compared. No order of the ring: ℤ/N has none compatible with its operations.

Detailed Description

template<auto N>
struct dedekind::morphologies::Modular< N >

\(\mathbb{Z}/N\mathbb{Z}\) as a total finite cyclic ring.

cyclic__The_Modular_Species

Relocated from dedekind.category:species into dedekind.morphologies:cyclic to live alongside the IsCyclic concept and its sibling carriers. The type satisfies both morphologies::IsCyclic (via the Domain / generator() / successor() member API) and dedekind::category::IsCyclicGroup<Modular<N>, std::plus<Modular<N>>> (via the trait specialisations below).

Member Typedef Documentation

wide_type

template<auto N>

using dedekind::morphologies::Modular< N >::wide_type

Initial value:

 
      std::conditional_t<(N == 1 ||
                          static_cast<std::uint64_t>(N - 1) <=
                              std::numeric_limits<std::uint64_t>::max() /
                                  static_cast<std::uint64_t>(N - 1)),
                         unsigned long long, unsigned __int128>

Member Function Documentation

inverse()

template<auto N>

Modular dedekind::morphologies::Modular< N >::inverse ( ) const

inlineconstexpr

The multiplicative inverse on a prime modulus, \(a^{N-2}\) (Fermat): ℤ/p is the field 𝔽ₚ.

Returns

the element whose product with this one is 1.

Exceptions

std::domain_error

for 0, which has no inverse.

Friends And Related Symbol Documentation

operator/

template<auto N>

friend Modular operator/ ( Modular< N > a, Modular< N > b )

friend

Division on a prime modulus: multiplication by the inverse.

Returns

\(a \cdot b^{-1}\).

Exceptions

std::domain_error

when b is 0.

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The documentation for this struct was generated from the following file:

• src/main/modules/dedekind/morphologies/cyclic.cppm