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

Class SimpleFieldExtension<E extends IUnivariatePolynomial<E>>

Constructor Summary

Method Summary

Methods inherited from class cc.redberry.rings.ARing

Methods inherited from class java.lang.Object

Methods inherited from interface java.util.Comparator

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

Methods inherited from interface java.lang.Iterable

Methods inherited from interface cc.redberry.rings.Ring

Constructor Detail

SimpleFieldExtension

Method Detail

isInTheBaseField

generator

degree

getMinimalPolynomial

getMinimalPolynomialRef

norm

conjugatesProduct

trace

normOfPolynomial

normOfPolynomial

minimalPolynomial

asMultipleExtension

nVariables

factory

isEuclideanRing

cardinality

characteristic

shouldReduceFast

add

subtract

multiply

negate

addMutable

subtractMutable

multiplyMutable

negateMutable

reciprocal

factor

getZero

getOne

isZero

isOne

valueOf

valueOfBigInteger

valueOf

copy

createArray

createArray2d

createArray2d

compare

randomElement

variable

parse

equals

hashCode

toString

toString

toString