Represents an element of a cyclotomic field. More...

#include <maths/cyclotomic.h>

Inheritance diagram for regina::Cyclotomic:

Public Types
using	Coefficient = Rational
	The type of each coefficient of the polynomial that is used to store a field element. More...

Public Member Functions
	Cyclotomic ()
	Creates an uninitialised field element. More...

	Cyclotomic (size_t field)
	Creates the zero element of the given cyclotomic field. More...

	Cyclotomic (size_t field, int value)
	Creates the given integer element within the given cyclotomic field. More...

	Cyclotomic (size_t field, const Rational &value)
	Creates the given rational element within the given cyclotomic field. More...

	Cyclotomic (const Cyclotomic &value)
	Creates a copy of the given field element, within the same cyclotomic field. More...

	Cyclotomic (Cyclotomic &&value) noexcept
	Moves the contents of the given field element to this new field element. More...

template<typename iterator >
	Cyclotomic (size_t field, iterator begin, iterator end)
	Creates a new field element from the given sequence of coefficients. More...

	Cyclotomic (size_t field, std::initializer_list< Rational > coefficients)
	Creates a new field element from a hard-coded sequence of coefficients. More...

	~Cyclotomic ()
	Destroys this field element. More...

void	init (size_t field)
	Initialises this to be the zero element of the given cyclotomic field. More...

size_t	field () const
	Returns the order n of the underlying cyclotomic field to which this element belongs. More...

size_t	degree () const
	Returns the degree of the polynomial that defines the underlying cyclotomic field. More...

const Rational &	operator[] (size_t exp) const
	Returns an individual rational coefficient of the polynomial representation of this field element. More...

Rational &	operator[] (size_t exp)
	Offers access to an individual rational coefficient of the polynomial representation of this field element. More...

Polynomial< Rational >	polynomial () const
	Returns the full polynomial representation of this field element. More...

std::complex< double >	evaluate (size_t whichRoot=1) const
	Returns the value of this cyclotomic field element as a complex number. More...

bool	operator== (const Cyclotomic &rhs) const
	Tests whether or not this and the given argument are the same element of the same cyclotomic field. More...

bool	operator!= (const Cyclotomic &rhs) const
	Tests whether or not this and the given argument are the same element of the same cyclotomic field. More...

Cyclotomic &	operator= (const Cyclotomic &value)
	Sets this to a copy of the given field element. More...

Cyclotomic &	operator= (Cyclotomic &&value) noexcept
	Moves the contents of the given field element to this field element. More...

Cyclotomic &	operator= (const Rational &scalar)
	Sets this field element to the given rational. More...

void	swap (Cyclotomic &other) noexcept
	Swaps the contents of this and the given field element. More...

void	negate ()
	Negates this field element. More...

void	invert ()
	Inverts this field element. More...

Cyclotomic	inverse () const
	Returns the inverse of this field element. More...

Cyclotomic &	operator*= (const Rational &scalar)
	Multiplies this field element by the given rational. More...

Cyclotomic &	operator/= (const Rational &scalar)
	Divides this field element by the given rational. More...

Cyclotomic &	operator+= (const Cyclotomic &other)
	Adds the given field element to this. More...

Cyclotomic &	operator-= (const Cyclotomic &other)
	Subtracts the given field element from this. More...

Cyclotomic &	operator*= (const Cyclotomic &other)
	Multiplies this by the given field element. More...

Cyclotomic &	operator/= (const Cyclotomic &other)
	Divides this by the given field element. More...

void	writeTextShort (std::ostream &out, bool utf8=false, const char *variable=nullptr) const
	Writes this field element to the given output stream, using the given variable name instead of `x`. More...

std::string	str (const char *variable) const
	Returns this field element as a human-readable string, using the given variable name instead of `x`. More...

std::string	utf8 (const char *variable) const
	Returns this field element as a human-readable string using unicode characters, using the given variable name instead of `x`. More...

void	writeTextLong (std::ostream &out) const
	A default implementation for detailed output. More...

std::string	str () const
	Returns a short text representation of this object. More...

std::string	utf8 () const
	Returns a short text representation of this object using unicode characters. More...

std::string	detail () const
	Returns a detailed text representation of this object. More...

Static Public Member Functions
static const Polynomial< Integer > &	cyclotomic (size_t n)
	Returns the nth cyclotomic polynomial `Φ_n`. More...

Friends
Cyclotomic	operator+ (const Cyclotomic &lhs, const Cyclotomic &rhs)
	Adds the two given cyclotomic field elements. More...

Cyclotomic	operator- (const Cyclotomic &lhs, const Cyclotomic &rhs)
	Subtracts the two given cyclotomic field elements. More...

Cyclotomic	operator* (const Cyclotomic &lhs, const Cyclotomic &rhs)
	Multiplies the two given cyclotomic field elements. More...

Detailed Description

Represents an element of a cyclotomic field.

The cyclotomic field of order n extends the rationals with a primitive nth root of unity. This is isomorphic to the polynomial field ℚ[x]/Φ_n, where Φ_n is the nth cyclotomic polynomial.

Using this isomorphism, each element of the cyclotomic field can be uniquely represented as a rational polynomial of degree strictly less than deg(Φ_n) = φ(n), where φ denotes Euler's totient function. This class stores field elements using such a polynomial representation, and does not store complex numbers directly. If you require the complex value of a field element (as a floating point approximation), you can call evaluate().

Each object of this class stores both the value of the field element and the order n of the underlying field. This means that you can freely work with elements of different fields simultaneously, though of course most operations (such as addition, multplication and so on) require all operands to belong to the same field.

This class requires that the order n is strictly positive.

This class implements C++ move semantics and adheres to the C++ Swappable requirement. It is designed to avoid deep copies wherever possible, even when passing or returning objects by value.

Although this class makes use of global data in its implementation, all of its methods are thread-safe.

Member Typedef Documentation

◆ Coefficient

using regina::Cyclotomic::Coefficient = Rational

The type of each coefficient of the polynomial that is used to store a field element.

Constructor & Destructor Documentation

◆ Cyclotomic() [1/8]

regina::Cyclotomic::Cyclotomic ( )

inline

Creates an uninitialised field element.

This element must be initialised using either init() or the assignment operator before it can be used.

The underlying cyclotomic field is not yet known; this will also be specified during the call to init() or the assignment operator.

◆ Cyclotomic() [2/8]

regina::Cyclotomic::Cyclotomic ( size_t field )

inlineexplicit

Creates the zero element of the given cyclotomic field.

Parameters

field the order of the underlying cyclotomic field; this must be strictly positive.

◆ Cyclotomic() [3/8]

regina::Cyclotomic::Cyclotomic	(	size_t	field,
		int	value
	)

inline

Creates the given integer element within the given cyclotomic field.

The polynomial representation of this element will simply be an integer constant.

Parameters

field	the order of the underlying cyclotomic field; this must be strictly positive.
value	the value of this element; that is, the integer constant.

◆ Cyclotomic() [4/8]

regina::Cyclotomic::Cyclotomic	(	size_t	field,
		const Rational &	value
	)

inline

Creates the given rational element within the given cyclotomic field.

The polynomial representation of this element will simply be a rational constant.

Parameters

field	the order of the underlying cyclotomic field; this must be strictly positive.
value	the value of this element; that is, the rational constant.

◆ Cyclotomic() [5/8]

regina::Cyclotomic::Cyclotomic ( const Cyclotomic & value )

inline

Creates a copy of the given field element, within the same cyclotomic field.

This constructor induces a deep copy of value.

Parameters

value the field element to copy.

◆ Cyclotomic() [6/8]

regina::Cyclotomic::Cyclotomic ( Cyclotomic && value )

inlinenoexcept

Moves the contents of the given field element to this new field element.

This is a fast (constant time) operation.

The element that was passed (value) will no longer be usable.

Parameters

value the field element to move.

◆ Cyclotomic() [7/8]

template<typename iterator >

regina::Cyclotomic::Cyclotomic	(	size_t	field,
		iterator	begin,
		iterator	end
	)

inline

Creates a new field element from the given sequence of coefficients.

The coefficients should describe the field element's polynomial representation, and should be given in order from the constant coefficient upwards. See operator[] for details on what this polynomial representation means.

There should be at most deg(Φ_n) = φ(n) coefficients in the list, where n is the given order of the underlying field; any missing coefficients are assumed to be zero. In particular, an empty sequence is allowed (and represents the zero field element).

Precondition: Rationals can be assigned values from dereferenced iterators of type iterator.

Python: Instead of a pair of iterators, this routine takes a python list of coefficients.

Parameters

field	the order of the underlying cyclotomic field; this must be strictly positive.
begin	the beginning of a sequence of at most `φ(n)` coefficients, as described above.
end	a past-the-end iterator indicating the end of the sequence of coefficients.

◆ Cyclotomic() [8/8]

regina::Cyclotomic::Cyclotomic	(	size_t	field,
		std::initializer_list< Rational >	coefficients
	)

inline

Creates a new field element from a hard-coded sequence of coefficients.

The coefficients should describe the field element's polynomial representation, and should be given in order from the constant coefficient upwards. See operator[] for details on what this polynomial representation means.

There should be at most deg(Φ_n) = φ(n) coefficients in the list, where n is the given order of the underlying field; any missing coefficients are assumed to be zero. In particular, an empty sequence is allowed (and represents the zero field element).

Python: Not present. Instead, use the Python constructor that takes a list of coefficients (which need not be constant).

Parameters

field	the order of the underlying cyclotomic field; this must be strictly positive.
coefficients	a sequence of at most `φ(n)` coefficients, as described above.

◆ ~Cyclotomic()

regina::Cyclotomic::~Cyclotomic ( )

inline

Destroys this field element.

This is safe even if the field element was never initialised.

Member Function Documentation

◆ cyclotomic()

static const Polynomial< Integer > & regina::Cyclotomic::cyclotomic ( size_t n )

static

Returns the nth cyclotomic polynomial Φ_n.

Cyclotomic polynomials are cached after they are computed, and so after the first call to cyclotomic(n), all subsequent calls with the same value of n will be essentially instantaneous.

Although it queries and manipulates a global cache, this routine is thread-safe.

Precondition: The given integer n must be strictly positive.

Python: Since Python exposes the class Polynomial<Rational> but not Polynomial<Integer>, this routine will convert the result to a Polynomial<Rational> (and will therefore return by value, not by reference).

Parameters

n	indicates which cyclotomic polynomial to return.

Returns: the cyclotomic polynomial Φ_n.

◆ degree()

size_t regina::Cyclotomic::degree ( ) const

inline

Returns the degree of the polynomial that defines the underlying cyclotomic field.

This is the degree of the cyclotomic polynomial Φ_n, and also the value of Euler's totient function φ(n), where n is the order of the field as returned by field().

A value of zero indicates that this field element has not yet been initialised (for instance, it was created using the default constructor).

Python: This is also used to implement the Python special method len().

Returns: the degree of the polynomial that defines the underlying field.

◆ detail()

std::string regina::Output< Cyclotomic , supportsUtf8 >::detail ( ) const

inherited

Returns a detailed text representation of this object.

This text may span many lines, and should provide the user with all the information they could want. It should be human-readable, should not contain extremely long lines (which cause problems for users reading the output in a terminal), and should end with a final newline. There are no restrictions on the underlying character set.

Returns: a detailed text representation of this object.

◆ evaluate()

std::complex< double > regina::Cyclotomic::evaluate ( size_t whichRoot = 1 ) const

Returns the value of this cyclotomic field element as a complex number.

The evaluation depends upon which primitive root of unity is used to build the underlying cyclotomic field of order n. This ambiguity is resolved as follows.

Suppose the polynomial representation of this field element in ℚ[x]/Φ_n (as described in the Cyclotomic class notes) is f(x). Then the evaluation of this field element will be f(ρ), where ρ is the nth root of unity ρ = exp(2πi × k/n), and where k is the argument whichRoot as passed to this routine.

Precondition: The argument whichRoot is coprime to n (the order of the underlying cyclotomic field).; This field element has been initialised (either through a non-default constructor, an assignment operator, or by calling init()).

Warning: This routine uses floating point arithmetic, and so the value that it returns is subject to the usual floating point error.

Parameters

whichRoot indicates which root of unity will be used to convert the polynomial representation of this field element into a complex number.

Returns: a floating-point approximation of this cyclotomic field element as a complex number.

◆ field()

size_t regina::Cyclotomic::field ( ) const

inline

Returns the order n of the underlying cyclotomic field to which this element belongs.

A value of zero indicates that this field element has not yet been initialised (for instance, it was created using the default constructor).

Returns: the order of the underlying cyclotomic field.

◆ init()

void regina::Cyclotomic::init ( size_t field )

inline

Initialises this to be the zero element of the given cyclotomic field.

This is safe even if this element was previously initialised as an element of a different field - all prior information about this field element will be safely discarded.

Parameters

field the order of the cyclotomic field to which this field element will now belong; this must be strictly positive.

◆ inverse()

Cyclotomic regina::Cyclotomic::inverse ( ) const

Returns the inverse of this field element.

This field element is not changed.

Precondition: This field element has already been initialised (and so it already has specified an underlying cyclotomic field).; This field element is non-zero.

◆ invert()

void regina::Cyclotomic::invert ( )

Inverts this field element.

This field element is changed directly.

Precondition: This field element has already been initialised (and so it already has specified an underlying cyclotomic field).; This field element is non-zero.

◆ negate()

void regina::Cyclotomic::negate ( )

inline

Negates this field element.

This field element is changed directly.

◆ operator!=()

bool regina::Cyclotomic::operator!= ( const Cyclotomic & rhs ) const

inline

Tests whether or not this and the given argument are the same element of the same cyclotomic field.

If this and rhs have different underlying fields then this test will always return true (indicating that the elements are not equal), even if they take the same numerical value when evaluated as complex numbers.

If either this or rhs have not been initialised (typically because they were created using the default constructor), then this comparison will return true. If both field elements have not been initialised, then this comparison will return false.

Parameters

rhs	the value to compare with this.

Returns: false if this and rhs are the same element of the same cyclotomic field, or true if they are not.

◆ operator*=() [1/2]

Cyclotomic & regina::Cyclotomic::operator*= ( const Cyclotomic & other )

Multiplies this by the given field element.

Precondition: The argument other belongs to the same cyclotomic field as this.

Parameters

other the field element to multiply this by.

Returns: a reference to this field element.

◆ operator*=() [2/2]

Cyclotomic & regina::Cyclotomic::operator*= ( const Rational & scalar )

inline

Multiplies this field element by the given rational.

This has the effect of multiplying the polynomial representation by a scalar constant.

Parameters

scalar the rational to multiply this by.

Returns: a reference to this field element.

◆ operator+=()

Cyclotomic & regina::Cyclotomic::operator+= ( const Cyclotomic & other )

inline

Adds the given field element to this.

Precondition: The argument other belongs to the same cyclotomic field as this.

Parameters

other the field element to add to this.

Returns: a reference to this field element.

◆ operator-=()

Cyclotomic & regina::Cyclotomic::operator-= ( const Cyclotomic & other )

inline

Subtracts the given field element from this.

Precondition: The argument other belongs to the same cyclotomic field as this.

Parameters

other the field element to subtract from this.

Returns: a reference to this field element.

◆ operator/=() [1/2]

Cyclotomic & regina::Cyclotomic::operator/= ( const Cyclotomic & other )

inline

Divides this by the given field element.

Precondition: The argument other is non-zero.; The argument other belongs to the same cyclotomic field as this.

Parameters

other the field element to divide this by.

Returns: a reference to this field element.

◆ operator/=() [2/2]

Cyclotomic & regina::Cyclotomic::operator/= ( const Rational & scalar )

inline

Divides this field element by the given rational.

This has the effect of dividing the polynomial representation by a scalar constant.

Precondition: The given rational is non-zero.

Parameters

scalar the rational to divide this by.

Returns: a reference to this field element.

◆ operator=() [1/3]

Cyclotomic & regina::Cyclotomic::operator= ( const Cyclotomic & value )

inline

Sets this to a copy of the given field element.

This assignment operator is safe even if this and value belong to different cyclotomic fields, or if this and/or value has not yet been initialised. The underlying field for this element will simply be changed to match the underlying field for value, and all old information stored for this element (if any) will be safely discarded. If value is uninitialised then this field element will become uninitialised also.

This operator induces a deep copy of value.

Parameters

value the new value to assign to this field element.

Returns: a reference to this field element.

◆ operator=() [2/3]

Cyclotomic & regina::Cyclotomic::operator= ( const Rational & scalar )

inline

Sets this field element to the given rational.

The underlying cyclotomic field will be left unchanged.

The polynomial representation for this field element will simply be a constant.

Precondition: This field element has already been initialised (and so it already has specified an underlying cyclotomic field).

Parameters

scalar the new rational value of this field element.

Returns: a reference to this field element.

◆ operator=() [3/3]

Cyclotomic & regina::Cyclotomic::operator= ( Cyclotomic && value )

inlinenoexcept

Moves the contents of the given field element to this field element.

This is a fast (constant time) operation.

This assignment operator is safe even if this and value belong to different cyclotomic fields, or if this and/or value has not yet been initialised. The underlying field for this element will simply be changed to match the underlying field for value, and all old information stored for this element (if any) will be safely discarded. If value is uninitialised then this field element will become uninitialised also.

The element that was passed (value) will no longer be usable.

Parameters

value the field element to move.

Returns: a reference to this field element.

◆ operator==()

bool regina::Cyclotomic::operator== ( const Cyclotomic & rhs ) const

inline

Tests whether or not this and the given argument are the same element of the same cyclotomic field.

If this and rhs have different underlying fields then this test will always return false, even if they take the same numerical value when evaluated as complex numbers.

If either this or rhs have not been initialised (typically because they were created using the default constructor), then this comparison will return false. If both field elements have not been initialised, then this comparison will return true.

Parameters

rhs	the value to compare with this.

Returns: true if and only if this and rhs are the same element of the same cyclotomic field.

◆ operator[]() [1/2]

Rational & regina::Cyclotomic::operator[] ( size_t exp )

inline

Offers access to an individual rational coefficient of the polynomial representation of this field element.

The polynomial representation expresses this field element as a member of ℚ[x]/Φ_n, using a rational polynomial of degree strictly less than deg(Φ_n) = φ(n); that is, strictly less than the value returned by degree(). See the Cyclotomic class notes for further details.

In particular, for a field element e, the operator e[i] will give access to the coefficient of x^i in this polynomial representation.

This routine returns a non-constant reference: you can use this to directly edit the coefficients (and therefore the value of the field element). Note that there is also a constant (read-only) variant of this routine.

Parameters

exp	indicates which coefficient to access; this must be between 0 and degree()-1 inclusive.

Returns: a reference to the corresponding rational coefficient.

◆ operator[]() [2/2]

const Rational & regina::Cyclotomic::operator[] ( size_t exp ) const

inline

Returns an individual rational coefficient of the polynomial representation of this field element.

The polynomial representation expresses this field element as a member of ℚ[x]/Φ_n, using a rational polynomial of degree strictly less than deg(Φ_n) = φ(n); that is, strictly less than the value returned by degree(). See the Cyclotomic class notes for further details.

In particular, for a field element e, the operator e[i] will return the coefficient of x^i in this polynomial representation.

This is a constant (read-only) routine; note that there is a non-constant (read-write) variant of this routine also.

Parameters

exp	indicates which coefficient to return; this must be between 0 and degree()-1 inclusive.

Returns: a constant reference to the corresponding rational coefficient.

◆ polynomial()

Polynomial< Rational > regina::Cyclotomic::polynomial ( ) const

inline

Returns the full polynomial representation of this field element.

The polynomial representation expresses this field element as a member of ℚ[x]/Φ_n, using a rational polynomial of degree strictly less than deg(Φ_n) = φ(n); that is, strictly less than the value returned by degree(). See the Cyclotomic class notes for further details.

Precondition: This field element has been initialised (either through a non-default constructor, an assignment operator, or by calling init()).

Returns: the full polynomial representation of this field element.

◆ str() [1/2]

std::string regina::Output< Cyclotomic , supportsUtf8 >::str ( ) const

inherited

Returns a short text representation of this object.

This text should be human-readable, should use plain ASCII characters where possible, and should not contain any newlines.

Within these limits, this short text ouptut should be as information-rich as possible, since in most cases this forms the basis for the Python __str__() and __repr__() functions.

Python: The Python "stringification" function __str__() will use precisely this function, and for most classes the Python __repr__() function will incorporate this into its output.

Returns: a short text representation of this object.

◆ str() [2/2]

std::string regina::Cyclotomic::str ( const char * variable ) const

inline

Returns this field element as a human-readable string, using the given variable name instead of x.

The field element will be written using its rational polynomial representation. The underlying field will not be indicated in the output, since this is often already understood. If required, it can be accessed by calling c.field().

Note: There is also the usual variant of str() which takes no arguments; that variant is inherited from the Output class.

Parameters

variable the symbol to use for the polynomial variable. This may be null, in which case the default variable x will be used.

Returns: this field element as a human-readable string.

◆ swap()

void regina::Cyclotomic::swap ( Cyclotomic & other )

inlinenoexcept

Swaps the contents of this and the given field element.

This is a fast (constant time) operation.

This and the given field element do not need to belong to the same cyclotomic field, and indeed one or both of them may be uninitialised. The underlying fields (if different) will be swapped accordingly.

Parameters

other the field element whose contents should be swapped with this.

◆ utf8() [1/2]

std::string regina::Output< Cyclotomic , supportsUtf8 >::utf8 ( ) const

inherited

Returns a short text representation of this object using unicode characters.

Like str(), this text should be human-readable, should not contain any newlines, and (within these constraints) should be as information-rich as is reasonable.

Unlike str(), this function may use unicode characters to make the output more pleasant to read. The string that is returned will be encoded in UTF-8.

Returns: a short text representation of this object.

◆ utf8() [2/2]

std::string regina::Cyclotomic::utf8 ( const char * variable ) const

inline

Returns this field element as a human-readable string using unicode characters, using the given variable name instead of x.

The field element will be written using its rational polynomial representation. The underlying field will not be indicated in the output, since this is often already understood. If required, it can be accessed by calling c.field().

This is similar to the output from str(), except that it uses unicode characters to make the output more pleasant to read. In particular, it makes use of superscript digits for exponents.

The string is encoded in UTF-8.

Note: There is also the usual variant of utf8() which takes no arguments; that variant is inherited from the Output class.

Parameters

variable the symbol to use for the polynomial variable. This may be null, in which case the default variable x will be used.

Returns: this field element as a unicode-enabled human-readable string.

◆ writeTextLong()

void regina::ShortOutput< Cyclotomic , supportsUtf8 >::writeTextLong ( std::ostream & out ) const

inlineinherited

A default implementation for detailed output.

This routine simply calls T::writeTextShort() and appends a final newline.

Python: Not present. Instead you can call detail() from the subclass T, which returns this output as a string.

Parameters

out	the output stream to which to write.

◆ writeTextShort()

void regina::Cyclotomic::writeTextShort	(	std::ostream &	out,
		bool	utf8 = `false`,
		const char *	variable = `nullptr`
	)		const

Writes this field element to the given output stream, using the given variable name instead of x.

The field element will be written using its rational polynomial representation. The underlying field will not be indicated in the output, since this is often already understood. If required, it can be accessed by calling c.field().

If utf8 is passed as true then unicode superscript characters will be used for exponents; these will be encoded using UTF-8. This will make the output nicer, but will require more complex fonts to be available on the user's machine.

Python: Not present. Use str() or utf8() instead.

Parameters

out	the output stream to which to write.
utf8	`true` if unicode superscript characters may be used.
variable	the symbol to use for the polynomial variable. This may be `null`, in which case the default variable `x` will be used.

Friends And Related Function Documentation

◆ operator*

Cyclotomic operator*	(	const Cyclotomic &	lhs,
		const Cyclotomic &	rhs
	)

friend

Multiplies the two given cyclotomic field elements.

Precondition: Both arguments belong to the same cyclotomic field.

Parameters

lhs	the first field element to multiply.
rhs	the second field element to multiply.

Returns: the product of both field elements.

◆ operator+

Cyclotomic operator+	(	const Cyclotomic &	lhs,
		const Cyclotomic &	rhs
	)

friend

Adds the two given cyclotomic field elements.

Precondition: Both arguments belong to the same cyclotomic field.

Parameters

lhs	the first field element to add.
rhs	the second field element to add.

Returns: the sum of both field elements.

◆ operator-

Cyclotomic operator-	(	const Cyclotomic &	lhs,
		const Cyclotomic &	rhs
	)

friend

Subtracts the two given cyclotomic field elements.

Precondition: Both arguments belong to the same cyclotomic field.

Parameters

lhs	the field element to subtract from.
rhs	the field element to subtract.

Returns: the first field element minus the second.

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

maths/cyclotomic.h

Public Types

Public Member Functions

Static Public Member Functions

Friends

Detailed Description

Member Typedef Documentation

◆ Coefficient

Constructor & Destructor Documentation

◆ Cyclotomic() [1/8]

◆ Cyclotomic() [2/8]

◆ Cyclotomic() [3/8]

◆ Cyclotomic() [4/8]

◆ Cyclotomic() [5/8]

◆ Cyclotomic() [6/8]

◆ Cyclotomic() [7/8]

◆ Cyclotomic() [8/8]

◆ ~Cyclotomic()

Member Function Documentation

◆ cyclotomic()

◆ degree()

◆ detail()

◆ evaluate()

◆ field()

◆ init()

◆ inverse()

◆ invert()

◆ negate()

◆ operator!=()

◆ operator*=() [1/2]

◆ operator*=() [2/2]

◆ operator+=()

◆ operator-=()

◆ operator/=() [1/2]

◆ operator/=() [2/2]

◆ operator=() [1/3]

◆ operator=() [2/3]

◆ operator=() [3/3]

◆ operator==()

◆ operator[]() [1/2]

◆ operator[]() [2/2]

◆ polynomial()

◆ str() [1/2]

◆ str() [2/2]

◆ swap()

◆ utf8() [1/2]

◆ utf8() [2/2]

◆ writeTextLong()

◆ writeTextShort()

Friends And Related Function Documentation

◆ operator*

◆ operator+

◆ operator-