MultipleFieldExtension (rings 2.5 API)

java.lang.Object
- cc.redberry.rings.ImageRing<sPoly,mPoly>
- - cc.redberry.rings.poly.MultipleFieldExtension<Term,mPoly,sPoly>

All Implemented Interfaces:

IParser<mPoly>, Stringifiable<mPoly>, IPolynomialRing<mPoly>, Ring<mPoly>, Serializable, Iterable<mPoly>, Comparator<mPoly>
```
public class MultipleFieldExtension<Term extends AMonomial<Term>,mPoly extends AMultivariatePolynomial<Term,mPoly>,sPoly extends IUnivariatePolynomial<sPoly>>
extends ImageRing<sPoly,mPoly>
implements IPolynomialRing<mPoly>
```
Multiple field extension F(α_1, α_2, ..., α_N). Elements are represented as multivariate polynomials over (α_1, α_2, ..., α_N); internally all arithmetic operations are performed by switching to appropriate simple field extension F(γ) (accessible via getSimpleExtension() method) where γ is some primitive element (accessible via getPrimitiveElement() method) computed automatically. Representation of generating algebraic elements α_i as elements of this simple extension can be obtained via getGeneratorRep(int) method. Originally, generators are represented by their minimal polynomials over F(α_1, α_2, ..., α_i) ("tower" representation). To construct multiple field extensions one should use #mkMultipleExtension(...) and joinAlgebraicElement(UnivariatePolynomial).

Since:

2.5

See Also:

mkMultipleExtension(IUnivariatePolynomial), mkMultipleExtension(IUnivariatePolynomial...), mkSplittingField(IUnivariatePolynomial), Rings.MultipleFieldExtension(IUnivariatePolynomial[]), Rings.SplittingField(IUnivariatePolynomial), Serialized Form

Field Summary
- Fields inherited from class cc.redberry.rings.ImageRing
  imageFunc, inverseFunc, ring

Constructor Summary

Constructors
Constructor and Description
`MultipleFieldExtension(MultipleFieldExtension<Term,mPoly,sPoly>[] tower, UnivariatePolynomial<mPoly>[] minimalPolynomialsOfGenerators, mPoly primitiveElement, sPoly[] generatorsReps, SimpleFieldExtension<sPoly> simpleExtension)`

Method Summary

All Methods Static Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`mPoly`	`copy(mPoly element)` Makes a deep copy of the specified element (for immutable instances the same reference returned).
`int`	`degree()` Returns the degree of this filed extension (that is the degree of primitive element)
`boolean`	`equals(Object o)`
`mPoly[]`	`extendedGCD(mPoly a, mPoly b)` Returns array of `[gcd(a,b), s, t]` such that `s * a + t * b = gcd(a, b)`
`mPoly`	`factorial(long num)` Gives a product of {@code valueOf(1) * valueOf(2) * ....
`mPoly`	`factory()` Factory polynomial
`mPoly`	`gcd(Iterable<mPoly> elements)` Returns greatest common divisor of specified elements
`mPoly`	`gcd(mPoly... elements)` Returns greatest common divisor of specified elements
`mPoly`	`gcd(mPoly a, mPoly b)` Returns the greatest common divisor of two elements
`UnivariatePolynomial<mPoly>`	`getGeneratorMinimalPoly(int iGenerator)` Returns minimal polynomial corresponding to i-th generator.
`sPoly`	`getGeneratorRep(int iGenerator)` Returns representation of i-th generator as element of simple field extension generated by primitive element `getPrimitiveElement()`
`sPoly[]`	`getGeneratorReps()` Returns representation of generators as elements of simple field extension generated by primitive element `getPrimitiveElement()`
`mPoly`	`getOne()` Returns unit element of this ring (one)
`mPoly`	`getPrimitiveElement()` Returns the primitive element of this multiple field extension
`SimpleFieldExtension<sPoly>`	`getSimpleExtension()` Returns the isomorphic simple field extension generated by `getPrimitiveElement()`
`MultipleFieldExtension<Term,mPoly,sPoly>`	`getSubExtension(int i)` Returns the i-th extension from the tower
`sPoly`	`getUnivariateFactory()`
`mPoly`	`getZero()` Returns zero element of this ring
`int`	`hashCode()`
`boolean`	`isOne(mPoly element)` Tests whether specified element is one (exactly)
`boolean`	`isUnit(mPoly element)` Tests whether specified element is a ring unit
`boolean`	`isZero(mPoly element)` Tests whether specified element is zero
`MultipleFieldExtension<Term,mPoly,sPoly>`	`joinAlgebraicElement(sPoly minimalPoly)` Adds algebraic element given by its minimal polynomial (not checked that it is irreducible) to this.
`MultipleFieldExtension<Term,mPoly,sPoly>`	`joinAlgebraicElement(UnivariatePolynomial<mPoly> algebraicElement)` Adds algebraic element given by its minimal polynomial (not checked that it is irreducible) to this.
`MultipleFieldExtension<Term,mPoly,sPoly>`	`joinRedundantElement(mPoly element)` Adds algebraic element given by its minimal polynomial (not checked that it is irreducible) to this.
`mPoly`	`lcm(mPoly a, mPoly b)` Returns the least common multiple of two elements
`static <Term extends AMonomial<Term>,mPoly extends AMultivariatePolynomial<Term,mPoly>,sPoly extends IUnivariatePolynomial<sPoly>> MultipleFieldExtension<Term,mPoly,sPoly>`	`mkMultipleExtension(SimpleFieldExtension<sPoly> ext)`
`static <Term extends AMonomial<Term>,mPoly extends AMultivariatePolynomial<Term,mPoly>,sPoly extends IUnivariatePolynomial<sPoly>> MultipleFieldExtension<Term,mPoly,sPoly>`	`mkMultipleExtension(sPoly... minimalPolynomials)` Creates multiple field extension `F(α_1, α_2, ..., α_i)` where `α_i` are specified by their minimal polynomials over F.
`static <Term extends AMonomial<Term>,mPoly extends AMultivariatePolynomial<Term,mPoly>,sPoly extends IUnivariatePolynomial<sPoly>> MultipleFieldExtension<Term,mPoly,sPoly>`	`mkMultipleExtension(sPoly a)`
`static <Term extends AMonomial<Term>,mPoly extends AMultivariatePolynomial<Term,mPoly>,sPoly extends IUnivariatePolynomial<sPoly>> MultipleFieldExtension<Term,mPoly,sPoly>`	`mkSplittingField(sPoly poly)` Constructs splitting field for a given polynomial.
`int`	`nVariables()` Number of polynomial variables
`int`	`signum(mPoly element)` Returns -1 if `element < 0`, 0 if `element == 0` and 1 if `element > 0`, where comparison is specified by `Comparator.compare(Object, Object)`
`String`	`toString()`
`String`	`toString(IStringifier stringifier)` convert this to string with the use of stringifier
`String`	`toString(String... variables)`
`mPoly`	`valueOf(long val)` Returns ring element associated with specified `long`
`mPoly`	`valueOfBigInteger(BigInteger val)` Returns ring element associated with specified integer
`mPoly`	`variable(int variable)` Creates poly representing a single specified variable

Methods inherited from class cc.redberry.rings.ImageRing
abs, add, add, cardinality, characteristic, compare, decrement, divideAndRemainder, factor, factorSquareFree, image, image, increment, inverse, inverse, isEuclideanRing, isField, isPerfectPower, iterator, multiply, multiply, negate, parse, perfectPowerBase, perfectPowerExponent, pow, pow, pow, quotient, randomElement, reciprocal, remainder, subtract, valueOf

Methods inherited from class java.lang.Object
clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Methods inherited from interface cc.redberry.rings.poly.IPolynomialRing
mkCoder, parse

Methods inherited from interface cc.redberry.rings.Ring
abs, add, add, addMutable, cardinality, characteristic, createArray, createArray, createArray, createArray, createArray2d, createArray2d, createZeroesArray, createZeroesArray2d, decrement, divideAndRemainder, divideExact, divideExactMutable, divideOrNull, factor, factorSquareFree, fillZeros, firstBezoutCoefficient, getNegativeOne, increment, isEuclideanRing, isField, isFinite, isFiniteField, isMinusOne, isPerfectPower, isUnitOrZero, iterator, lcm, lcm, max, min, multiply, multiply, multiply, multiply, multiplyMutable, negate, negateMutable, parse, perfectPowerBase, perfectPowerExponent, pow, pow, pow, quotient, randomElement, randomElement, randomElementTree, randomElementTree, randomNonZeroElement, reciprocal, remainder, setToValueOf, subtract, subtractMutable, valueOf, valueOf

Methods inherited from interface java.util.Comparator
compare, comparing, comparing, comparingDouble, comparingInt, comparingLong, naturalOrder, nullsFirst, nullsLast, reversed, reverseOrder, thenComparing, thenComparing, thenComparing, thenComparingDouble, thenComparingInt, thenComparingLong

Methods inherited from interface java.lang.Iterable
forEach, spliterator

- Constructor Detail
  - MultipleFieldExtension
```
public MultipleFieldExtension(MultipleFieldExtension<Term,mPoly,sPoly>[] tower,
                              UnivariatePolynomial<mPoly>[] minimalPolynomialsOfGenerators,
                              mPoly primitiveElement,
                              sPoly[] generatorsReps,
                              SimpleFieldExtension<sPoly> simpleExtension)
```
- Method Detail
  - nVariables
```
public int nVariables()
```
    Description copied from interface: IPolynomialRing
    
    Number of polynomial variables
    
    Specified by:
    
    nVariables in interface IPolynomialRing<mPoly extends AMultivariatePolynomial<Term,mPoly>>
  - factory
```
public mPoly factory()
```
    Description copied from interface: IPolynomialRing
    
    Factory polynomial
    
    Specified by:
    
    factory in interface IPolynomialRing<mPoly extends AMultivariatePolynomial<Term,mPoly>>
  - variable
```
public mPoly variable(int variable)
```
    Description copied from interface: IPolynomialRing
    
    Creates poly representing a single specified variable
    
    Specified by:
    
    variable in interface IPolynomialRing<mPoly extends AMultivariatePolynomial<Term,mPoly>>
  - getUnivariateFactory
```
public sPoly getUnivariateFactory()
```
  - getPrimitiveElement
```
public mPoly getPrimitiveElement()
```
    Returns the primitive element of this multiple field extension
  - degree
```
public int degree()
```
    Returns the degree of this filed extension (that is the degree of primitive element)
  - getSimpleExtension
```
public SimpleFieldExtension<sPoly> getSimpleExtension()
```
    Returns the isomorphic simple field extension generated by getPrimitiveElement()
  - getGeneratorMinimalPoly
```
public UnivariatePolynomial<mPoly> getGeneratorMinimalPoly(int iGenerator)
```
    Returns minimal polynomial corresponding to i-th generator.
  - getSubExtension
```
public MultipleFieldExtension<Term,mPoly,sPoly> getSubExtension(int i)
```
    Returns the i-th extension from the tower
  - getGeneratorRep
```
public sPoly getGeneratorRep(int iGenerator)
```
    Returns representation of i-th generator as element of simple field extension generated by primitive element getPrimitiveElement()
  - getGeneratorReps
```
public sPoly[] getGeneratorReps()
```
    Returns representation of generators as elements of simple field extension generated by primitive element getPrimitiveElement()
  - joinAlgebraicElement
```
public MultipleFieldExtension<Term,mPoly,sPoly> joinAlgebraicElement(UnivariatePolynomial<mPoly> algebraicElement)
```
    Adds algebraic element given by its minimal polynomial (not checked that it is irreducible) to this.
  - joinAlgebraicElement
```
public MultipleFieldExtension<Term,mPoly,sPoly> joinAlgebraicElement(sPoly minimalPoly)
```
    Adds algebraic element given by its minimal polynomial (not checked that it is irreducible) to this.
  - joinRedundantElement
```
public MultipleFieldExtension<Term,mPoly,sPoly> joinRedundantElement(mPoly element)
```
    Adds algebraic element given by its minimal polynomial (not checked that it is irreducible) to this.
  - valueOf
```
public mPoly valueOf(long val)
```
    Description copied from interface: Ring
    
    Returns ring element associated with specified long
    
    Specified by:
    
    valueOf in interface Ring<mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Overrides:
    
    valueOf in class ImageRing<sPoly extends IUnivariatePolynomial<sPoly>,mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Parameters:
    
    val - machine integer
    
    Returns:
    
    ring element associated with specified long
  - valueOfBigInteger
```
public mPoly valueOfBigInteger(BigInteger val)
```
    Description copied from interface: Ring
    
    Returns ring element associated with specified integer
    
    Specified by:
    
    valueOfBigInteger in interface Ring<mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Overrides:
    
    valueOfBigInteger in class ImageRing<sPoly extends IUnivariatePolynomial<sPoly>,mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Parameters:
    
    val - integer
    
    Returns:
    
    ring element associated with specified integer
  - getZero
```
public mPoly getZero()
```
    Description copied from interface: Ring
    
    Returns zero element of this ring
    
    Specified by:
    
    getZero in interface Ring<mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Overrides:
    
    getZero in class ImageRing<sPoly extends IUnivariatePolynomial<sPoly>,mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Returns:
    
    0
  - getOne
```
public mPoly getOne()
```
    Description copied from interface: Ring
    
    Returns unit element of this ring (one)
    
    Specified by:
    
    getOne in interface Ring<mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Overrides:
    
    getOne in class ImageRing<sPoly extends IUnivariatePolynomial<sPoly>,mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Returns:
    
    1
  - copy
```
public mPoly copy(mPoly 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<mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Overrides:
    
    copy in class ImageRing<sPoly extends IUnivariatePolynomial<sPoly>,mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Parameters:
    
    element - the element
    
    Returns:
    
    deep copy of specified element
  - isZero
```
public boolean isZero(mPoly element)
```
    Description copied from interface: Ring
    
    Tests whether specified element is zero
    
    Specified by:
    
    isZero in interface Ring<mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Overrides:
    
    isZero in class ImageRing<sPoly extends IUnivariatePolynomial<sPoly>,mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Parameters:
    
    element - the ring element
    
    Returns:
    
    whether specified element is zero
  - isOne
```
public boolean isOne(mPoly element)
```
    Description copied from interface: Ring
    
    Tests whether specified element is one (exactly)
    
    Specified by:
    
    isOne in interface Ring<mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Overrides:
    
    isOne in class ImageRing<sPoly extends IUnivariatePolynomial<sPoly>,mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Parameters:
    
    element - the ring element
    
    Returns:
    
    whether specified element is exactly one
    
    See Also:
    
    Ring.isUnit(Object)
  - isUnit
```
public boolean isUnit(mPoly element)
```
    Description copied from interface: Ring
    
    Tests whether specified element is a ring unit
    
    Specified by:
    
    isUnit in interface Ring<mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Overrides:
    
    isUnit in class ImageRing<sPoly extends IUnivariatePolynomial<sPoly>,mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Parameters:
    
    element - the ring element
    
    Returns:
    
    whether specified element is a ring unit
    
    See Also:
    
    Ring.isOne(Object)
  - gcd
```
public mPoly gcd(mPoly a,
                 mPoly b)
```
    Description copied from interface: Ring
    
    Returns the greatest common divisor of two elements
    
    Specified by:
    
    gcd in interface Ring<mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Overrides:
    
    gcd in class ImageRing<sPoly extends IUnivariatePolynomial<sPoly>,mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Parameters:
    
    a - the first element
    
    b - the second element
    
    Returns:
    
    gcd
  - extendedGCD
```
public mPoly[] extendedGCD(mPoly a,
                           mPoly b)
```
    Description copied from interface: Ring
    
    Returns array of [gcd(a,b), s, t] such that s * a + t * b = gcd(a, b)
    
    Specified by:
    
    extendedGCD in interface Ring<mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Overrides:
    
    extendedGCD in class ImageRing<sPoly extends IUnivariatePolynomial<sPoly>,mPoly extends AMultivariatePolynomial<Term,mPoly>>
  - lcm
```
public mPoly lcm(mPoly a,
                 mPoly b)
```
    Description copied from interface: Ring
    
    Returns the least common multiple of two elements
    
    Specified by:
    
    lcm in interface Ring<mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Overrides:
    
    lcm in class ImageRing<sPoly extends IUnivariatePolynomial<sPoly>,mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Parameters:
    
    a - the first element
    
    b - the second element
    
    Returns:
    
    lcm
  - gcd
```
public mPoly gcd(mPoly... elements)
```
    Description copied from interface: Ring
    
    Returns greatest common divisor of specified elements
    
    Specified by:
    
    gcd in interface Ring<mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Overrides:
    
    gcd in class ImageRing<sPoly extends IUnivariatePolynomial<sPoly>,mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Parameters:
    
    elements - the elements
    
    Returns:
    
    gcd
  - gcd
```
public mPoly gcd(Iterable<mPoly> elements)
```
    Description copied from interface: Ring
    
    Returns greatest common divisor of specified elements
    
    Specified by:
    
    gcd in interface Ring<mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Overrides:
    
    gcd in class ImageRing<sPoly extends IUnivariatePolynomial<sPoly>,mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Parameters:
    
    elements - the elements
    
    Returns:
    
    gcd
  - signum
```
public int signum(mPoly element)
```
    Description copied from interface: Ring
    
    Returns -1 if element < 0, 0 if element == 0 and 1 if element > 0, where comparison is specified by Comparator.compare(Object, Object)
    
    Specified by:
    
    signum in interface IPolynomialRing<mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Specified by:
    
    signum in interface Ring<mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Overrides:
    
    signum in class ImageRing<sPoly extends IUnivariatePolynomial<sPoly>,mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Parameters:
    
    element - the element
    
    Returns:
    
    -1 if element < 0, 0 if element == 0 and 1 otherwise
  - factorial
```
public mPoly factorial(long num)
```
    Description copied from interface: Ring
    
    Gives a product of valueOf(1) * valueOf(2) * .... * valueOf(num)
    
    Specified by:
    
    factorial in interface Ring<mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Overrides:
    
    factorial in class ImageRing<sPoly extends IUnivariatePolynomial<sPoly>,mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Parameters:
    
    num - the number
    
    Returns:
    
    valueOf(1) * valueOf(2) * .... * valueOf(num)
  - equals
```
public boolean equals(Object o)
```
    Specified by:
    
    equals in interface Comparator<mPoly extends AMultivariatePolynomial<Term,mPoly>>
    
    Overrides:
    
    equals in class ImageRing<sPoly extends IUnivariatePolynomial<sPoly>,mPoly extends AMultivariatePolynomial<Term,mPoly>>
  - hashCode
```
public int hashCode()
```
    Overrides:
    
    hashCode in class ImageRing<sPoly extends IUnivariatePolynomial<sPoly>,mPoly extends AMultivariatePolynomial<Term,mPoly>>
  - toString
```
public String toString(IStringifier stringifier)
```
    Description copied from interface: Stringifiable
    
    convert this to string with the use of stringifier
    
    Specified by:
    
    toString in interface Stringifiable<mPoly extends AMultivariatePolynomial<Term,mPoly>>
  - toString
```
public String toString(String... variables)
```
  - toString
```
public String toString()
```
    Overrides:
    
    toString in class Object
  - mkMultipleExtension
```
public static <Term extends AMonomial<Term>,mPoly extends AMultivariatePolynomial<Term,mPoly>,sPoly extends IUnivariatePolynomial<sPoly>> MultipleFieldExtension<Term,mPoly,sPoly> mkMultipleExtension(sPoly a)
```
  - mkMultipleExtension
```
public static <Term extends AMonomial<Term>,mPoly extends AMultivariatePolynomial<Term,mPoly>,sPoly extends IUnivariatePolynomial<sPoly>> MultipleFieldExtension<Term,mPoly,sPoly> mkMultipleExtension(SimpleFieldExtension<sPoly> ext)
```
  - mkMultipleExtension
```
public static <Term extends AMonomial<Term>,mPoly extends AMultivariatePolynomial<Term,mPoly>,sPoly extends IUnivariatePolynomial<sPoly>> MultipleFieldExtension<Term,mPoly,sPoly> mkMultipleExtension(sPoly... minimalPolynomials)
```
    Creates multiple field extension F(α_1, α_2, ..., α_i) where α_i are specified by their minimal polynomials over F.
    NOTE: it is not tested that specified minimal polynomials are irreducible
    
    Parameters:
    
    minimalPolynomials - minimal polynomials of algebraic elements
  - mkSplittingField
```
public static <Term extends AMonomial<Term>,mPoly extends AMultivariatePolynomial<Term,mPoly>,sPoly extends IUnivariatePolynomial<sPoly>> MultipleFieldExtension<Term,mPoly,sPoly> mkSplittingField(sPoly poly)
```
    Constructs splitting field for a given polynomial.

Class MultipleFieldExtension<Term extends AMonomial<Term>,mPoly extends AMultivariatePolynomial<Term,mPoly>,sPoly extends IUnivariatePolynomial<sPoly>>

Field Summary

Fields inherited from class cc.redberry.rings.ImageRing

Constructor Summary

Method Summary

Methods inherited from class cc.redberry.rings.ImageRing

Methods inherited from class java.lang.Object

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

Methods inherited from interface cc.redberry.rings.Ring

Methods inherited from interface java.util.Comparator

Methods inherited from interface java.lang.Iterable

Constructor Detail

MultipleFieldExtension

Method Detail

nVariables

factory

variable

getUnivariateFactory

getPrimitiveElement

degree

getSimpleExtension

getGeneratorMinimalPoly

getSubExtension

getGeneratorRep

getGeneratorReps

joinAlgebraicElement

joinAlgebraicElement

joinRedundantElement

valueOf

valueOfBigInteger

getZero

getOne

copy

isZero

isOne

isUnit

gcd

extendedGCD

lcm

gcd

gcd

signum

factorial

equals

hashCode

toString

toString

toString

mkMultipleExtension

mkMultipleExtension

mkMultipleExtension

mkSplittingField