Section8.11

]>
Book Contents Previous Section 8.10 Solution of Differential Equations Next Section 8.12 Primary Decomposition of Ideals Book Index

## 8.11 Finite Fields

A finite field (also called a Galois field) is a finite algebraic structure where one can add, multiply and divide under the same laws (for example, commutativity, associativity or distributivity) as apply to the rational, real or complex numbers. Unlike those three fields, for any finite field there exists a positive prime integer p, called the characteristic, such that p x=0 for any element x in the finite field. In fact, the number of elements in a finite field is a power of the characteristic and for each prime p and positive integer n there exists exactly one finite field with p n elements, up to isomorphism. For more information about the algebraic structure and properties of finite fields, see, for example,
S. Lang, Algebra, Second Edition, New York: Addison-Wesley Publishing Company, Inc., 1984, ISBN 0 201 05487 6;
or
R. Lidl, H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and Its Applications, Vol. 20, Cambridge: Cambridge Univ. Press, 1983, ISBN 0 521 30240 4.

When n=1, the field has p elements and is called a prime field, discussed in the next section. There are several ways of implementing extensions of finite fields, and Axiom provides quite a bit of freedom to allow you to choose the one that is best for your application. Moreover, we provide operations for converting among the different representations of extensions and different extensions of a single field. Finally, note that you usually need to package-call operations from finite fields if the operations do not take as an argument an object of the field. See ugTypesPkgCall for more information on package-calling.

### 8.11.1 Modular Arithmetic and Prime Fields

finite field Galois:field field:finite:prime field:prime field:Galois prime field modular arithmetic arithmetic:modular

Let n be a positive integer. It is well known that you can get the same result if you perform addition, subtraction or multiplication of integers and then take the remainder on dividing by n as if you had first done such remaindering on the operands, performed the arithmetic and then (if necessary) done remaindering again. This allows us to speak of arithmetic modulo n or, more simply mod n.

In Axiom, you use IntegerMod to do such arithmetic.

(a,b) : IntegerMod 12
Type: Void
(a, b) := (16, 7)

7

Type: IntegerMod 12
[a - b, a * b]

[9, 4]

Type: List IntegerMod 12

If n is not prime, there is only a limited notion of reciprocals and division.

a / b
There are 12 exposed and 13 unexposed library operations named /
having 2 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op /
to learn more about the available operations. Perhaps
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named /
with argument type(s)
IntegerMod 12
IntegerMod 12
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
recip a

"failed"

Type: Union("failed",...)

Here 7 and 12 are relatively prime, so 7 has a multiplicative inverse modulo 12.

recip b

7

Type: Union(IntegerMod 12,...)

If we take n to be a prime number p, then taking inverses and, therefore, division are generally defined.

Use PrimeField instead of IntegerMod for n prime.

c : PrimeField 11 := 8

8

Type: PrimeField 11
inv c

7

Type: PrimeField 11

You can also use 1/c and c**(-1) for the inverse of c.

9/c

8

Type: PrimeField 11

PrimeField (abbreviation PF) checks if its argument is prime when you try to use an operation from it. If you know the argument is prime (particularly if it is large), InnerPrimeField (abbreviation IPF) assumes the argument has already been verified to be prime. If you do use a number that is not prime, you will eventually get an error message, most likely a division by zero message. For computer science applications, the most important finite fields are PrimeField 2 and its extensions.

In the following examples, we work with the finite field with p=101 elements.

GF101 := PF 101

PrimeField101

Type: Domain

Like many domains in Axiom, finite fields provide an operation for returning a random element of the domain.

x := random()$GF101

8

Type: PrimeField 101
y : GF101 := 37

37

Type: PrimeField 101
z := x/y

63

Type: PrimeField 101
z * y - x

0

Type: PrimeField 101

The element 2 is a primitive element of this field, primitive element element:primitive

pe := primitiveElement()$GF101

2

Type: PrimeField 101

in the sense that its powers enumerate all nonzero elements.

[pe**i for i in 0..99]

[1, 2, 4, 8, 16, 32, 64, 27, 54, 7, 14, 28, 56, 11, 22, 44, 88, 75, 49, 98, 95, 89, 77, 53, 5, 10, 20, 40, 80, 59, 17, 34, 68, 35, 70, 39, 78, 55, 9, 18, 36, 72, 43, 86, 71, 41, 82, 63, 25, 50, 100, 99, 97, 93, 85, 69, 37, 74, 47, 94, 87, 73, 45, 90, 79, 57, 13, 26, 52, 3, 6, 12, 24, 48, 96, 91, 81, 61, 21, 42, 84, 67, 33, 66, 31, 62, 23, 46, 92, 83, 65, 29, 58, 15, 30, 60, 19, 38, 76, 51]

Type: List PrimeField 101

If every nonzero element is a power of a primitive element, how do you determine what the exponent is? Use discrete logarithm discreteLog. logarithm:discrete

ex := discreteLog(y)

56

Type: PositiveInteger
pe ** ex

37

Type: PrimeField 101

The order of a nonzero element x is the smallest positive integer t such x t=1.

order y

25

Type: PositiveInteger

The order of a primitive element is the defining p-1.

order pe

100

Type: PositiveInteger

### 8.11.2 Extensions of Finite Fields

finite field field:finite:extension of

When you want to work with an extension of a finite field in Axiom, you have three choices to make:

1. Do you want to generate an extension of the prime field (for example, PrimeField 2) or an extension of a given field?
2. Do you want to use a representation that is particularly efficient for multiplication, exponentiation and addition but uses a lot of computer memory (a representation that models the cyclic group structure of the multiplicative group of the field extension and uses a Zech logarithm table), one that Zech logarithm uses a normal basis for the vector space structure of the field extension, or one that performs arithmetic modulo an irreducible polynomial? The cyclic group representation is only usable up to ``medium'' (relative to your machine's performance) sized fields. If the field is large and the normal basis is relatively simple, the normal basis representation is more efficient for exponentiation than the irreducible polynomial representation.
3. Do you want to provide a polynomial explicitly, a root of which ``generates'' the extension in one of the three senses in (2), or do you wish to have the polynomial generated for you?

This illustrates one of the most important features of Axiom: you can choose exactly the right data-type and representation to suit your application best.

We first tell you what domain constructors to use for each case above, and then give some examples.

Constructors that automatically generate extensions of the prime field:
FiniteField
FiniteFieldCyclicGroup
FiniteFieldNormalBasis

Constructors that generate extensions of an arbitrary field:
FiniteFieldExtension
FiniteFieldExtensionByPolynomial
FiniteFieldCyclicGroupExtension
FiniteFieldCyclicGroupExtensionByPolynomial
FiniteFieldNormalBasisExtension
FiniteFieldNormalBasisExtensionByPolynomial

Constructors that use a cyclic group representation:
FiniteFieldCyclicGroup
FiniteFieldCyclicGroupExtension
FiniteFieldCyclicGroupExtensionByPolynomial

Constructors that use a normal basis representation:
FiniteFieldNormalBasis
FiniteFieldNormalBasisExtension
FiniteFieldNormalBasisExtensionByPolynomial

Constructors that use an irreducible modulus polynomial representation:
FiniteField
FiniteFieldExtension
FiniteFieldExtensionByPolynomial

Constructors that generate a polynomial for you:
FiniteField
FiniteFieldExtension
FiniteFieldCyclicGroup
FiniteFieldCyclicGroupExtension
FiniteFieldNormalBasis
FiniteFieldNormalBasisExtension

Constructors for which you provide a polynomial:
FiniteFieldExtensionByPolynomial
FiniteFieldCyclicGroupExtensionByPolynomial
FiniteFieldNormalBasisExtensionByPolynomial

These constructors are discussed in the following sections where we collect together descriptions of extension fields that have the same underlying representation.For more information on the implementation aspects of finite fields, see J. Grabmeier, A. Scheerhorn, Finite Fields in AXIOM, Technical Report, IBM Heidelberg Scientific Center, 1992.

If you don't really care about all this detail, just use FiniteField. As your knowledge of your application and its Axiom implementation grows, you can come back and choose an alternative constructor that may improve the efficiency of your code. Note that the exported operations are almost the same for all constructors of finite field extensions and include the operations exported by PrimeField.

### 8.11.3 Irreducible Modulus Polynomial Representations

All finite field extension constructors discussed in this finite field section field:finite:extension of use a representation that performs arithmetic with univariate (one-variable) polynomials modulo an irreducible polynomial. This polynomial may be given explicitly by you or automatically generated. The ground field may be the prime field or one you specify. See ugxProblemFiniteExtensionFinite for general information about finite field extensions.

For FiniteField (abbreviation FF) you provide a prime number p and an extension degree n. This degree can be 1.

Axiom uses the prime field PrimeField(p), here PrimeField 2, and it chooses an irreducible polynomial of degree n, here 12, over the ground field.

GF4096 := FF(2,12);
Type: Domain

The objects in the generated field extension are polynomials of degree at most n-1 with coefficients in the prime field. The polynomial indeterminate is automatically chosen by Axiom and is typically something like %A or %D. These (strange) variables are only for output display; there are several ways to construct elements of this field.

The operation index enumerates the elements of the field extension and accepts as argument the integers from 1 to p n.

The expression index(p) always gives the indeterminate.

a := index(2)$GF4096

%A

Type: FiniteField(2,12)

You can build polynomials in a and calculate in GF4096.

b := a**12 - a**5 + a

%A 5+%A 3+%A+1

Type: FiniteField(2,12)
b ** 1000

%A 10+%A 9+%A 7+%A 5+%A 4+%A 3+%A

Type: FiniteField(2,12)
c := a/b

%A 11+%A 8+%A 7+%A 5+%A 4+%A 3+%A 2

Type: FiniteField(2,12)

Among the available operations are norm and trace.

norm c

1

Type: PrimeField 2
trace c

0

Type: PrimeField 2

Since any nonzero element is a power of a primitive element, how do we discover what the exponent is?

The operation discreteLog calculates discrete logarithm the exponent and, logarithm:discrete if it is called with only one argument, always refers to the primitive element returned by primitiveElement.

dL := discreteLog a

1729

Type: PositiveInteger
g ** dL

g 1729

Type: Polynomial Integer

FiniteFieldExtension (abbreviation FFX) is similar to FiniteField except that the ground-field for FiniteFieldExtension is arbitrary and chosen by you.

In case you select the prime field as ground field, there is essentially no difference between the constructed two finite field extensions.

GF16 := FF(2,4);
Type: Domain
GF4096 := FFX(GF16,3);
Type: Domain
r := (random()$GF4096) ** 20

(%B 2+1)%C 2+(%B 3+%B 2+1)%C+%B 3+%B 2+%B+1

Type: FiniteFieldExtension(FiniteField(2,4),3)
norm(r)

%B 2+%B

Type: FiniteField(2,4)

FiniteFieldExtensionByPolynomial (abbreviation FFP) is similar to FiniteField and FiniteFieldExtension but is more general.

GF4 := FF(2,2);
Type: Domain
f := nextIrreduciblePoly(random(6)$FFPOLY(GF4))$FFPOLY(GF4)

Type: Union(SparseUnivariatePolynomial FiniteField(2,2),...)

For FFP you choose both the ground field and the irreducible polynomial used in the representation. The degree of the extension is the degree of the polynomial.

GF4096 := FFP(GF4,f);
Type: Domain
discreteLog random()$GF4096

582

Type: PositiveInteger

### 8.11.4 Cyclic Group Representations

finite field field:finite:extension of

In every finite field there exist elements whose powers are all the nonzero elements of the field. Such an element is called a primitive element.

In FiniteFieldCyclicGroup (abbreviation FFCG) group:cyclic the nonzero elements are represented by the powers of a fixed primitive element:primitive element primitive element of the field (that is, a generator of its cyclic multiplicative group). Multiplication (and hence exponentiation) using this representation is easy. To do addition, we consider our primitive element as the root of a primitive polynomial (an irreducible polynomial whose roots are all primitive). See ugxProblemFiniteUtility for examples of how to compute such a polynomial.

To use FiniteFieldCyclicGroup you provide a prime number and an extension degree.

GF81 := FFCG(3,4);
Type: Domain

Axiom uses the prime field, here PrimeField 3, as the ground field and it chooses a primitive polynomial of degree n, here 4, over the prime field.

a := primitiveElement()$GF81

%F 1

Type: FiniteFieldCyclicGroup(3,4)

You can calculate in GF81.

b := a**12 - a**5 + a

%F 72

Type: FiniteFieldCyclicGroup(3,4)

In this representation of finite fields the discrete logarithm of an element can be seen directly in its output form.

b

%F 72

Type: FiniteFieldCyclicGroup(3,4)
discreteLog b

72

Type: PositiveInteger

FiniteFieldCyclicGroupExtension (abbreviation FFCGX) is similar to FiniteFieldCyclicGroup except that the ground field for FiniteFieldCyclicGroupExtension is arbitrary and chosen by you. In case you select the prime field as ground field, there is essentially no difference between the constructed two finite field extensions.

GF9 := FF(3,2);
Type: Domain
GF729 := FFCGX(GF9,3);
Type: Domain
r := (random()$GF729) ** 20

%H 420

Type: FiniteFieldCyclicGroupExtension(FiniteField(3,2),3)
trace(r)

0

Type: FiniteField(3,2)

FiniteFieldCyclicGroupExtensionByPolynomial (abbreviation FFCGP) is similar to FiniteFieldCyclicGroup and FiniteFieldCyclicGroupExtension but is more general. For FiniteFieldCyclicGroupExtensionByPolynomial you choose both the ground field and the irreducible polynomial used in the representation. The degree of the extension is the degree of the polynomial.

GF3 := PrimeField 3;
Type: Domain

We use a utility operation to generate an irreducible primitive polynomial (see ugxProblemFiniteUtility ). The polynomial has one variable that is ``anonymous'': it displays as a question mark.

f := createPrimitivePoly(4)$FFPOLY(GF3)

Type: SparseUnivariatePolynomial PrimeField 3
GF81 := FFCGP(GF3,f);
Type: Domain

Let's look at a random element from this field.

random()$GF81

%K 13

Type: FiniteFieldCyclicGroupExtensionByPolynomial(PrimeField 3,?**4+?+2)

### 8.11.5 Normal Basis Representations

finite field field:finite:extension of basis:normal normal basis

Let K be a finite extension of degree n of the finite field F and let F have q elements. An element x of K is said to be normal over F if the elements

1, x q, x q 2,…,x q n-1

form a basis of K as a vector space over F. Such a basis is called a normal basis.This agrees with the general definition of a normal basis because the n distinct powers of the automorphism x↦x q constitute the Galois group of K/F.

If x is normal over F, its minimal polynomial:minimal polynomial is also said to be normal over F. minimal polynomial There exist normal bases for all finite extensions of arbitrary finite fields.

In FiniteFieldNormalBasis (abbreviation FFNB), the elements of the finite field are represented by coordinate vectors with respect to a normal basis.

You provide a prime p and an extension degree n.

K := FFNB(3,8)

FiniteFieldNormalBasis(3, 8)

Type: Domain

Axiom uses the prime field PrimeField(p), here PrimeField 3, and it chooses a normal polynomial of degree n, here 8, over the ground field. The remainder class of the indeterminate is used as the normal element. The polynomial indeterminate is automatically chosen by Axiom and is typically something like %A or %D. These (strange) variables are only for output display; there are several ways to construct elements of this field. The output of the basis elements is something like %A q i.

a := normalElement()$K

%I

Type: FiniteFieldNormalBasis(3,8)

You can calculate in K using a.

b := a**12 - a**5 + a

2%I q 7+%I q 5+%I q

Type: FiniteFieldNormalBasis(3,8)

FiniteFieldNormalBasisExtension (abbreviation FFNBX) is similar to FiniteFieldNormalBasis except that the groundfield for FiniteFieldNormalBasisExtension is arbitrary and chosen by you. In case you select the prime field as ground field, there is essentially no difference between the constructed two finite field extensions.

GF9 := FFNB(3,2);
Type: Domain
GF729 := FFNBX(GF9,3);
Type: Domain
r := random()$GF729

2%K%L q

Type: FiniteFieldNormalBasisExtension(FiniteFieldNormalBasis(3,2),3)
r + r**3 + r**9 + r**27

2%K%L q 2+(2%K q+2%K)%L q+2%K q%L

Type: FiniteFieldNormalBasisExtension(FiniteFieldNormalBasis(3,2),3)

FiniteFieldNormalBasisExtensionByPolynomial (abbreviation FFNBP) is similar to FiniteFieldNormalBasis and FiniteFieldNormalBasisExtension but is more general. For FiniteFieldNormalBasisExtensionByPolynomial you choose both the ground field and the irreducible polynomial used in the representation. The degree of the extension is the degree of the polynomial.

GF3 := PrimeField 3;
Type: Domain

We use a utility operation to generate an irreducible normal polynomial (see ugxProblemFiniteUtility ). The polynomial has one variable that is ``anonymous'': it displays as a question mark.

f := createNormalPoly(4)$FFPOLY(GF3)

?4+2? 3+2

Type: SparseUnivariatePolynomial PrimeField 3
GF81 := FFNBP(GF3,f);
Type: Domain

Let's look at a random element from this field.

r := random()$GF81

%M q 2+2%M q+2%M

Type: FiniteFieldNormalBasisExtensionByPolynomial(PrimeField 3,?**4+2*?**3+2)
r * r**3 * r**9 * r**27

2%M q 3+2%M q 2+2%M q+2%M

Type: FiniteFieldNormalBasisExtensionByPolynomial(PrimeField 3,?**4+2*?**3+2)
norm r

2

Type: PrimeField 3

### 8.11.6 Conversion Operations for Finite Fields

field:finite:conversions

Let K be a finite field.

K := PrimeField 3

PrimeField3

Type: Domain

An extension field K m of degree m over K is a subfield of an extension field K n of degree n over K if and only if m divides n.

K n |  |
❘ |  |
K m | ⟺ | m∣n
❘ |  |
K |  |

FiniteFieldHomomorphisms provides conversion operations between different extensions of one fixed finite ground field and between different representations of these finite fields.

Let's choose m and n,

(m,n) := (4,8)

8

Type: PositiveInteger

build the field extensions,

Km := FiniteFieldExtension(K,m)

FiniteFieldExtension(PrimeField3,4)

Type: Domain

and pick two random elements from the smaller field.

Kn := FiniteFieldExtension(K,n)

FiniteFieldExtension(PrimeField3,8)

Type: Domain
a1 := random()$Km

2%A 3+%A 2

Type: FiniteFieldExtension(PrimeField 3,4)
b1 := random()$Km

%A 3+%A 2+2%A+1

Type: FiniteFieldExtension(PrimeField 3,4)

Since m divides n, K m is a subfield of K n.

a2 := a1 :: Kn

%B 4

Type: FiniteFieldExtension(PrimeField 3,8)

Therefore we can convert the elements of K m into elements of K n.

b2 := b1 :: Kn

2%B 6+2%B 4+%B 2+1

Type: FiniteFieldExtension(PrimeField 3,8)

To check this, let's do some arithmetic.

a1+b1 - ((a2+b2) :: Km)

0

Type: FiniteFieldExtension(PrimeField 3,4)
a1*b1 - ((a2*b2) :: Km)

0

Type: FiniteFieldExtension(PrimeField 3,4)

There are also conversions available for the situation, when K m and K n are represented in different ways (see ugxProblemFiniteExtensionFinite ). For example let's choose K m where the representation is 0 plus the cyclic multiplicative group and K n with a normal basis representation.

Km := FFCGX(K,m)

FiniteFieldCyclicGroupExtension(PrimeField3,4)

Type: Domain
Kn := FFNBX(K,n)

FiniteFieldNormalBasisExtension(PrimeField3,8)

Type: Domain
(a1,b1) := (random()$Km,random()$Km)

%C 13

Type: FiniteFieldCyclicGroupExtension(PrimeField 3,4)
a2 := a1 :: Kn

2%D q 6+2%D q 5+2%D q 4+2%D q 2+2%D q+2%D

Type: FiniteFieldNormalBasisExtension(PrimeField 3,8)
b2 := b1 :: Kn

2%D q 7+%D q 6+%D q 5+%D q 4+2%D q 3+%D q 2+%D q+%D

Type: FiniteFieldNormalBasisExtension(PrimeField 3,8)

Check the arithmetic again.

a1+b1 - ((a2+b2) :: Km)

0

Type: FiniteFieldCyclicGroupExtension(PrimeField 3,4)
a1*b1 - ((a2*b2) :: Km)

0

Type: FiniteFieldCyclicGroupExtension(PrimeField 3,4)

### 8.11.7 Utility Operations for Finite Fields

FiniteFieldPolynomialPackage (abbreviation FFPOLY) provides operations for generating, counting and testing polynomials over finite fields. Let's start with a couple of definitions:

- A polynomial is primitive if its roots are primitive polynomial:primitive elements in an extension of the coefficient field of degree equal to the degree of the polynomial.
- A polynomial is normal over its coefficient field polynomial:normal if its roots are linearly independent elements in an extension of the coefficient field of degree equal to the degree of the polynomial.

In what follows, many of the generated polynomials have one ``anonymous'' variable. This indeterminate is displayed as a question mark (``?'').

To fix ideas, let's use the field with five elements for the first few examples.

GF5 := PF 5;
Type: Domain

You can generate irreducible polynomials of any (positive) degree polynomial:irreducible (within the storage capabilities of the computer and your ability to wait) by using createIrreduciblePolycreateIrreduciblePolyFiniteFieldPolynomialPackage.

f := createIrreduciblePoly(8)$FFPOLY(GF5)

?8+?4+2

Type: SparseUnivariatePolynomial PrimeField 5

Does this polynomial have other important properties? Use primitive? to test whether it is a primitive polynomial.

primitive?(f)$FFPOLY(GF5)

false

Type: Boolean

Use normal? to test whether it is a normal polynomial.

normal?(f)$FFPOLY(GF5)

false

Type: Boolean

Note that this is actually a trivial case, because a normal polynomial of degree n must have a nonzero term of degree n-1. We will refer back to this later.

To get a primitive polynomial of degree 8 just issue this.

p := createPrimitivePoly(8)$FFPOLY(GF5)

?8+?3+?2+?+2

Type: SparseUnivariatePolynomial PrimeField 5
primitive?(p)$FFPOLY(GF5)

true

Type: Boolean

This polynomial is not normal,

normal?(p)$FFPOLY(GF5)

false

Type: Boolean

but if you want a normal one simply write this.

n := createNormalPoly(8)$FFPOLY(GF5)

?8+4? 7+?3+1

Type: SparseUnivariatePolynomial PrimeField 5

This polynomial is not primitive!

primitive?(n)$FFPOLY(GF5)

false

Type: Boolean

This could have been seen directly, as the constant term is 1 here, which is not a primitive element up to the factor ( -1) raised to the degree of the polynomial.Cf. Lidl, R. & Niederreiter, H., Finite Fields, Encycl. of Math. 20, (Addison-Wesley, 1983), p.90, Th. 3.18.

What about polynomials that are both primitive and normal? The existence of such a polynomial is by no means obvious. The existence of such polynomials is proved in Lenstra, H. W. & Schoof, R. J., Primitive Normal Bases for Finite Fields, Math. Comp. 48, 1987, pp. 217-231.

If you really need one use either createPrimitiveNormalPolycreatePrimitiveNormalPolyFiniteFieldPolynomialPackage or createNormalPrimitivePolycreateNormalPrimitivePolyFiniteFieldPolynomialPackage.

createPrimitiveNormalPoly(8)$FFPOLY(GF5)

?8+4? 7+2? 5+2

Type: SparseUnivariatePolynomial PrimeField 5

If you want to obtain additional polynomials of the various types above as given by the create... operations above, you can use the next... operations. For instance, nextIrreduciblePolynextIrreduciblePolyFiniteFieldPolynomialPackage yields the next monic irreducible polynomial with the same degree as the input polynomial. By ``next'' we mean ``next in a natural order using the terms and coefficients.'' This will become more clear in the following examples.

This is the field with five elements.

GF5 := PF 5;
Type: Domain

Our first example irreducible polynomial, say of degree 3, must be ``greater'' than this.

h := monomial(1,8)$SUP(GF5)

?8

Type: SparseUnivariatePolynomial PrimeField 5

You can generate it by doing this.

nh := nextIrreduciblePoly(h)$FFPOLY(GF5)

?8+2

Type: Union(SparseUnivariatePolynomial PrimeField 5,...)

Notice that this polynomial is not the same as the one createIrreduciblePolycreateIrreduciblePolyFiniteFieldPolynomialPackage.

createIrreduciblePoly(3)$FFPOLY(GF5)

?3+?+1

Type: SparseUnivariatePolynomial PrimeField 5

You can step through all irreducible polynomials of degree 8 over the field with 5 elements by repeatedly issuing this.

nh := nextIrreduciblePoly(nh)$FFPOLY(GF5)

?8+3

Type: Union(SparseUnivariatePolynomial PrimeField 5,...)

You could also ask for the total number of these.

numberOfIrreduciblePoly(5)$FFPOLY(GF5)

624

Type: PositiveInteger

We hope that ``natural order'' on polynomials is now clear: first we compare the number of monomials of two polynomials (``more'' is ``greater''); then, if necessary, the degrees of these monomials (lexicographically), and lastly their coefficients (also lexicographically, and using the operation lookup if our field is not a prime field). Also note that we make both polynomials monic before looking at the coefficients: multiplying either polynomial by a nonzero constant produces the same result.

The package FiniteFieldPolynomialPackage also provides similar operations for primitive and normal polynomials. With the exception of the number of primitive normal polynomials; we're not aware of any known formula for this.

numberOfPrimitivePoly(3)$FFPOLY(GF5)

20

Type: PositiveInteger

Take these,

m := monomial(1,1)$SUP(GF5)

?

Type: SparseUnivariatePolynomial PrimeField 5
f := m**3 + 4*m**2 + m + 2

?3+4? 2+?+2

Type: SparseUnivariatePolynomial PrimeField 5

and then we have:

f1 := nextPrimitivePoly(f)$FFPOLY(GF5)

?3+4? 2+4?+2

Type: Union(SparseUnivariatePolynomial PrimeField 5,...)

What happened?

nextPrimitivePoly(f1)$FFPOLY(GF5)

?3+2? 2+3

Type: Union(SparseUnivariatePolynomial PrimeField 5,...)

Well, for the ordering used in nextPrimitivePolynextPrimitivePolyFiniteFieldPolynomialPackage we use as first criterion a comparison of the constant terms of the polynomials. Analogously, in nextNormalPolynextNormalPolyFiniteFieldPolynomialPackage we first compare the monomials of degree 1 less than the degree of the polynomials (which is nonzero, by an earlier remark).

f := m**3 + m**2 + 4*m + 1

?3+?2+4?+1

Type: SparseUnivariatePolynomial PrimeField 5
f1 := nextNormalPoly(f)$FFPOLY(GF5)

?3+?2+4?+3

Type: Union(SparseUnivariatePolynomial PrimeField 5,...)
nextNormalPoly(f1)$FFPOLY(GF5)

?3+2? 2+1

Type: Union(SparseUnivariatePolynomial PrimeField 5,...)

We don't have to restrict ourselves to prime fields.

Let's consider, say, a field with 16 elements.

GF16 := FFX(FFX(PF 2,2),2);
Type: Domain

We can apply any of the operations described above.

createIrreduciblePoly(5)$FFPOLY(GF16)

?5+%G

Type: SparseUnivariatePolynomial FiniteFieldExtension(FiniteFieldExtension(PrimeField 2,2),2)

Axiom also provides operations for producing random polynomials of a given degree

random(5)$FFPOLY(GF16)

?5+(%F%G+1)?4+%F%G? 3+(%G+%F+1)?2+((%F+1)%G+%F)?+1

Type: SparseUnivariatePolynomial FiniteFieldExtension(FiniteFieldExtension(PrimeField 2,2),2)

or with degree between two given bounds.

random(3,9)$FFPOLY(GF16)

?3+(%F%G+1)?2+(%G+%F+1)?+1

Type: SparseUnivariatePolynomial FiniteFieldExtension(FiniteFieldExtension(PrimeField 2,2),2)

FiniteFieldPolynomialPackage2 (abbreviation FFPOLY2) exports an operation rootOfIrreduciblePoly for finding one root of an irreducible polynomial f polynomial:root of in an extension field of the coefficient field. The degree of the extension has to be a multiple of the degree of f. It is not checked whether f actually is irreducible.

To illustrate this operation, we fix a ground field GF

GF2 := PrimeField 2;
Type: Domain

and then an extension field.

F := FFX(GF2,12)

FiniteFieldExtension(PrimeField2,12)

Type: Domain

We construct an irreducible polynomial over GF2.

f := createIrreduciblePoly(6)$FFPOLY(GF2)

?6+?+1

Type: SparseUnivariatePolynomial PrimeField 2

We compute a root of f.

root := rootOfIrreduciblePoly(f)$FFPOLY2(F,GF2)

%H 11+%H 8+%H 7+%H 5+%H+1

Type: FiniteFieldExtension(PrimeField 2,12)

and check the result

eval(f, monomial(1,1)$SUP(F) = root)

0

Type: SparseUnivariatePolynomial FiniteFieldExtension(PrimeField 2,12)
Book Contents Previous Section 8.10 Solution of Differential Equations Next Section 8.12 Primary Decomposition of Ideals Book Index