Package cc.redberry.rings.poly

Class SimpleFieldExtension<E extends IUnivariatePolynomial<E>>

java.lang.Object

cc.redberry.rings.ARing<E>

cc.redberry.rings.poly.SimpleFieldExtension<E>

All Implemented Interfaces:

IParser<E>, Stringifiable<E>, IPolynomialRing<E>, Ring<E>, Serializable, Iterable<E>, Comparator<E>

Direct Known Subclasses:

AlgebraicNumberField, FiniteField

public abstract class SimpleFieldExtension<E extends IUnivariatePolynomial<E>>
extends ARing<E>
implements IPolynomialRing<E>

A simple field extension F(α) represented as a univariate quotient ring F[x]/<m(x)> where m(x) is the minimal polynomial of α. Elements of extension field are represented as univariate polynomials in α. To create simple field extensions one should use either FiniteField for extensions of finite fields or AlgebraicNumberField for extensions of rationals. See MultipleFieldExtension for implementation of multiple extensions.

Since:

2.5

See Also:

FiniteField, AlgebraicNumberField, Rings.SimpleFieldExtension(IUnivariatePolynomial), MultipleFieldExtension, Serialized Form

Constructor Summary

Constructors

Modifier	Constructor	Description
protected	SimpleFieldExtension(E minimalPoly)	Constructs a simple field extension F(α) generated by the algebraic number α with the specified minimal polynomial.

Method Summary

Modifier and Type	Method	Description
E	add(E a, E b)	Add two elements
E	addMutable(E a, E b)	Adds two elements and destroys the initial content of a.
<Term extends AMonomial<Term>,mPoly extends AMultivariatePolynomial<Term,mPoly>> MultipleFieldExtension<Term,mPoly,E>	asMultipleExtension()	Returns a view of this as a multiple field extension
BigInteger	cardinality()	Returns the number of elements in this ring (cardinality) or null if ring is infinite
BigInteger	characteristic()	Returns characteristic of this ring
int	compare(E o1, E o2)
E	conjugatesProduct(E element)	Gives the product of all conjugates of given element (except element itself), that is norm(element) / element
E	copy(E element)	Makes a deep copy of the specified element (for immutable instances the same reference returned).
E[]	createArray(int length)	Creates generic array of ring elements of specified length
E[][]	createArray2d(int length)	Creates 2d array of ring elements of specified length
E[][]	createArray2d(int m, int n)	Creates 2d array of ring elements of specified shape
int	degree()	Returns the degree of this filed extension (that is the degree of minimal polynomial)
boolean	equals(Object o)
FactorDecomposition<E>	factor(E element)	Factor specified element
E	factory()	Factory polynomial
E	generator()	Returns the generator element α of this field extension F(α)
E	getMinimalPolynomial()	Returns the minimal polynomial of the generator (that is the "modulo" polynomial p(x) of this field viewed as quotient field F[x]/<p(x)>)
E	getMinimalPolynomialRef()	INTERNAL
E	getOne()	Returns unit element of this ring (one)
E	getZero()	Returns zero element of this ring
int	hashCode()
boolean	isEuclideanRing()	Returns whether this ring is a Euclidean ring
boolean	isInTheBaseField(E element)	Returns whether the given element belongs to the base field
boolean	isOne(E element)	Tests whether specified element is one (exactly)
boolean	isZero(E element)	Tests whether specified element is zero
E	minimalPolynomial(E element)	Computes minimal polynomial of a given algebraic element
E	multiply(E a, E b)	Multiplies two elements
E	multiplyMutable(E a, E b)	Multiplies two elements and destroys the initial content of a
E	negate(E element)	Negates the given element
E	negateMutable(E element)	Negates the given element and destroys the initial content of element
E	norm(E element)	Gives the norm of field extension element (it is always belongs to the base field)
<MPoly extends AMultivariatePolynomial> MPoly	normOfPolynomial(MultivariatePolynomial<E> poly)	Gives the norm of multivariate polynomial over this field extension, which is always a polynomial with the coefficients from the base field.
E	normOfPolynomial(UnivariatePolynomial<E> poly)	Gives the norm of univariate polynomial over this field extension, which is always a polynomial with the coefficients from the base field
int	nVariables()	Number of polynomial variables
E	parse(String string)	Parse string into ring element
E	randomElement(org.apache.commons.math3.random.RandomGenerator rnd)	Returns a random element from this ring
E	reciprocal(E element)	Gives the inverse element element ^ (-1)
protected boolean	shouldReduceFast(int dividendDegree)	empiric to switch between fast and plain division
E	subtract(E a, E b)	Subtracts b from a
E	subtractMutable(E a, E b)	Subtracts b from a and destroys the initial content of a
String	toString()
String	toString(IStringifier<E> stringifier)	convert this to string with the use of stringifier
String	toString(String... variables)
E	trace(E element)	Gives the trace of field extension element (it is always belongs to the base field)
E	valueOf(long val)	Returns ring element associated with specified long
E	valueOf(E val)	Converts a value from other ring to this ring.
E	valueOfBigInteger(BigInteger val)	Returns ring element associated with specified integer
E	variable(int variable)	Creates poly representing a single specified variable

Methods inherited from class cc.redberry.rings.ARing

isPerfectPower, perfectPowerBase, perfectPowerExponent

Methods inherited from class java.lang.Object

clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Methods inherited from interface java.util.Comparator

reversed, thenComparing, thenComparing, thenComparing, thenComparingDouble, thenComparingInt, thenComparingLong

Methods inherited from interface cc.redberry.rings.poly.IPolynomialRing

mkCoder, parse, signum

Methods inherited from interface java.lang.Iterable

forEach, spliterator

Methods inherited from interface cc.redberry.rings.Ring

abs, add, createArray, createArray, createArray, createZeroesArray, createZeroesArray2d, decrement, divideAndRemainder, divideExact, divideExactMutable, divideOrNull, extendedGCD, factorial, factorSquareFree, fillZeros, firstBezoutCoefficient, gcd, gcd, gcd, getNegativeOne, increment, isField, isFinite, isFiniteField, isMinusOne, isPerfectPower, isUnit, isUnitOrZero, iterator, lcm, lcm, lcm, max, min, multiply, multiply, multiply, perfectPowerBase, perfectPowerExponent, pow, pow, pow, quotient, randomElement, randomElementTree, randomElementTree, randomNonZeroElement, remainder, setToValueOf, valueOf

Constructor Detail

SimpleFieldExtension

protected SimpleFieldExtension(E minimalPoly)

Constructs a simple field extension F(α) generated by the algebraic number α with the specified minimal polynomial.

NOTE: irreducibility test for the minimal polynomial is not performed here, use IrreduciblePolynomials.irreducibleQ(IUnivariatePolynomial) to test irreducibility.

Parameters:

minimalPoly - the minimal polynomial

Method Detail

isInTheBaseField

public boolean isInTheBaseField(E element)

Returns whether the given element belongs to the base field

generator

public E generator()

Returns the generator element α of this field extension F(α)

degree

public int degree()

Returns the degree of this filed extension (that is the degree of minimal polynomial)

getMinimalPolynomial

public E getMinimalPolynomial()

Returns the minimal polynomial of the generator (that is the "modulo" polynomial p(x) of this field viewed as quotient field F[x]/<p(x)>)

getMinimalPolynomialRef

public E getMinimalPolynomialRef()

INTERNAL

norm

public E norm(E element)

Gives the norm of field extension element (it is always belongs to the base field)

conjugatesProduct

public E conjugatesProduct(E element)

Gives the product of all conjugates of given element (except element itself), that is norm(element) / element

trace

public E trace(E element)

Gives the trace of field extension element (it is always belongs to the base field)

normOfPolynomial

public E normOfPolynomial(UnivariatePolynomial<E> poly)

Gives the norm of univariate polynomial over this field extension, which is always a polynomial with the coefficients from the base field

normOfPolynomial

public <MPoly extends AMultivariatePolynomial> MPoly normOfPolynomial(MultivariatePolynomial<E> poly)

Gives the norm of multivariate polynomial over this field extension, which is always a polynomial with the coefficients from the base field.

minimalPolynomial

public E minimalPolynomial(E element)

Computes minimal polynomial of a given algebraic element

asMultipleExtension

public <Term extends AMonomial<Term>,mPoly extends AMultivariatePolynomial<Term,mPoly>> MultipleFieldExtension<Term,mPoly,E> asMultipleExtension()

Returns a view of this as a multiple field extension

nVariables

public int nVariables()

Description copied from interface: IPolynomialRing

Number of polynomial variables

Specified by:

nVariables in interface IPolynomialRing<E extends IUnivariatePolynomial<E>>

factory

public E factory()

Description copied from interface: IPolynomialRing

Factory polynomial

Specified by:

factory in interface IPolynomialRing<E extends IUnivariatePolynomial<E>>

isEuclideanRing

public boolean isEuclideanRing()

Description copied from interface: Ring

Returns whether this ring is a Euclidean ring

Specified by:

isEuclideanRing in interface Ring<E extends IUnivariatePolynomial<E>>

Returns:

whether this ring is a Euclidean ring

cardinality

public BigInteger cardinality()

Description copied from interface: Ring

Returns the number of elements in this ring (cardinality) or null if ring is infinite

Specified by:

cardinality in interface Ring<E extends IUnivariatePolynomial<E>>

Returns:

the number of elements in this ring (cardinality) or null if ring is infinite

characteristic

public BigInteger characteristic()

Description copied from interface: Ring

Returns characteristic of this ring

Specified by:

characteristic in interface Ring<E extends IUnivariatePolynomial<E>>

Returns:

characteristic of this ring

shouldReduceFast

protected boolean shouldReduceFast(int dividendDegree)

empiric to switch between fast and plain division

add

public E add(E a,
             E b)

Description copied from interface: Ring

Add two elements

Specified by:

add in interface Ring<E extends IUnivariatePolynomial<E>>

Parameters:

a - the first element

b - the second element

Returns:

a + b

subtract

public E subtract(E a,
                  E b)

Description copied from interface: Ring

Subtracts b from a

Specified by:

subtract in interface Ring<E extends IUnivariatePolynomial<E>>

Parameters:

a - the first element

b - the second element

Returns:

a - b

multiply

public E multiply(E a,
                  E b)

Description copied from interface: Ring

Multiplies two elements

Specified by:

multiply in interface Ring<E extends IUnivariatePolynomial<E>>

Parameters:

a - the first element

b - the second element

Returns:

a * b

negate

public E negate(E element)

Description copied from interface: Ring

Negates the given element

Specified by:

negate in interface Ring<E extends IUnivariatePolynomial<E>>

Parameters:

element - the ring element

Returns:

-val

addMutable

public E addMutable(E a,
                    E b)

Description copied from interface: Ring

Adds two elements and destroys the initial content of a.

Specified by:

addMutable in interface Ring<E extends IUnivariatePolynomial<E>>

Parameters:

a - the first element (may be destroyed)

b - the second element

Returns:

a + b

subtractMutable

public E subtractMutable(E a,
                         E b)

Description copied from interface: Ring

Subtracts b from a and destroys the initial content of a

Specified by:

subtractMutable in interface Ring<E extends IUnivariatePolynomial<E>>

Parameters:

a - the first element (may be destroyed)

b - the second element

Returns:

a - b

multiplyMutable

public E multiplyMutable(E a,
                         E b)

Description copied from interface: Ring

Multiplies two elements and destroys the initial content of a

Specified by:

multiplyMutable in interface Ring<E extends IUnivariatePolynomial<E>>

Parameters:

a - the first element (may be destroyed)

b - the second element

Returns:

a * b

negateMutable

public E negateMutable(E element)

Description copied from interface: Ring

Negates the given element and destroys the initial content of element

Specified by:

negateMutable in interface Ring<E extends IUnivariatePolynomial<E>>

Parameters:

element - the ring element (may be destroyed)

Returns:

-element

reciprocal

public E reciprocal(E element)

Description copied from interface: Ring

Gives the inverse element element ^ (-1)

Specified by:

reciprocal in interface Ring<E extends IUnivariatePolynomial<E>>

Parameters:

element - the element

Returns:

element ^ (-1)

factor

public FactorDecomposition<E> factor(E element)

Description copied from interface: Ring

Factor specified element

Specified by:

factor in interface Ring<E extends IUnivariatePolynomial<E>>

getZero

public E getZero()

Description copied from interface: Ring

Returns zero element of this ring

Specified by:

getZero in interface Ring<E extends IUnivariatePolynomial<E>>

Returns:

0

getOne

public E getOne()

Description copied from interface: Ring

Returns unit element of this ring (one)

Specified by:

getOne in interface Ring<E extends IUnivariatePolynomial<E>>

Returns:

1

isZero

public boolean isZero(E element)

Description copied from interface: Ring

Tests whether specified element is zero

Specified by:

isZero in interface Ring<E extends IUnivariatePolynomial<E>>

Parameters:

element - the ring element

Returns:

whether specified element is zero

isOne

public boolean isOne(E element)

Description copied from interface: Ring

Tests whether specified element is one (exactly)

Specified by:

isOne in interface Ring<E extends IUnivariatePolynomial<E>>

Parameters:

element - the ring element

Returns:

whether specified element is exactly one

See Also:

Ring.isUnit(Object)

valueOf

public E valueOf(long val)

Description copied from interface: Ring

Returns ring element associated with specified long

Specified by:

valueOf in interface Ring<E extends IUnivariatePolynomial<E>>

Parameters:

val - machine integer

Returns:

ring element associated with specified long

valueOfBigInteger

public E valueOfBigInteger(BigInteger val)

Description copied from interface: Ring

Returns ring element associated with specified integer

Specified by:

valueOfBigInteger in interface Ring<E extends IUnivariatePolynomial<E>>

Parameters:

val - integer

Returns:

ring element associated with specified integer

valueOf

public E valueOf(E val)

Description copied from interface: Ring

Converts a value from other ring to this ring. The result is not guarantied to be a new instance (i.e. val == valueOf(val) is possible).

Specified by:

valueOf in interface Ring<E extends IUnivariatePolynomial<E>>

Parameters:

val - some element from any ring

Returns:

this ring element associated with specified val

copy

public E copy(E element)

Description copied from interface: Ring

Makes a deep copy of the specified element (for immutable instances the same reference returned).

Specified by:

copy in interface Ring<E extends IUnivariatePolynomial<E>>

Parameters:

element - the element

Returns:

deep copy of specified element

createArray

public E[] createArray(int length)

Description copied from interface: Ring

Creates generic array of ring elements of specified length

Specified by:

createArray in interface Ring<E extends IUnivariatePolynomial<E>>

Parameters:

length - array length

Returns:

array of ring elements of specified length

createArray2d

public E[][] createArray2d(int length)

Description copied from interface: Ring

Creates 2d array of ring elements of specified length

Specified by:

createArray2d in interface Ring<E extends IUnivariatePolynomial<E>>

Parameters:

length - array length

Returns:

2d array of ring elements of specified length

createArray2d

public E[][] createArray2d(int m,
                           int n)

Description copied from interface: Ring

Creates 2d array of ring elements of specified shape

Specified by:

createArray2d in interface Ring<E extends IUnivariatePolynomial<E>>

Parameters:

m - result length

n - length of each array in the result

Returns:

2d array E[m][n]

compare

public int compare(E o1,
                   E o2)

Specified by:

compare in interface Comparator<E extends IUnivariatePolynomial<E>>

randomElement

public E randomElement(org.apache.commons.math3.random.RandomGenerator rnd)

Description copied from interface: Ring

Returns a random element from this ring

Specified by:

randomElement in interface Ring<E extends IUnivariatePolynomial<E>>

Parameters:

rnd - the source of randomness

Returns:

random element from this ring

variable

public E variable(int variable)

Description copied from interface: IPolynomialRing

Creates poly representing a single specified variable

Specified by:

variable in interface IPolynomialRing<E extends IUnivariatePolynomial<E>>

parse

public E parse(String string)

Description copied from interface: Ring

Parse string into ring element

Specified by:

parse in interface IParser<E extends IUnivariatePolynomial<E>>

Specified by:

parse in interface Ring<E extends IUnivariatePolynomial<E>>

Parameters:

string - string

Returns:

ring element

See Also:

Coder

equals

public boolean equals(Object o)

Specified by:

equals in interface Comparator<E extends IUnivariatePolynomial<E>>

Overrides:

equals in class Object

hashCode

public int hashCode()

Overrides:

hashCode in class Object

toString

public String toString(IStringifier<E> stringifier)

Description copied from interface: Stringifiable

convert this to string with the use of stringifier

Specified by:

toString in interface Stringifiable<E extends IUnivariatePolynomial<E>>

toString

public String toString(String... variables)

toString

public String toString()

Overrides:

toString in class Object