CAS, Commutative Rings, Fraction Fields, Quotient Rings, Polynomial Rings, Galois Fields
Author KlausC
3 Stars
Updated Last
24 Days Ago
Started In
July 2019


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This software is the start of a computer algebra system specialized to discrete calculations in the area of integer numbers , modular arithmetic ℤ/m fractional , polynomials ℤ[x]. Also multivariate polynomials ℤ[x,y,...] and Galois fields GF(p^r) are supported.

This is not seen as a replacement of Nemo.jl or AbstractAlgebra.jl, which should be used for serious work. It is understood more like a sandbox to try out a simpler API.

It is important, that rings may be freely combined, for example (ℤ/p)[x] (polynomials over the quotient ring for a prime number p), Frac(ℤ[x]), the rational functions with integer coefficients, or GF(64)[:x], polynomials over the Galois field. The quotient rings include ideals, which are of major importance with multivariate polynomials.

The mentioned examples are elementary examples for ring structures. The can be liberately combined to fractional fields, quotient rings, and polynomials of previously defined structures.

So it is possible to work with rational functions, which are fractions of polynomials, where the polynomial coefficients are in ℤ/p, for example. The the current standard library we have modules Rational and Polynomial besides the numeric subtypes of Number and some support for modular calculations with integers.

The original motivation for writing this piece of sofware, when I tried to handle polynomials over a quotient ring. There was no obvious way of embedding my ring elements into the Julia language and for example exploit polynomial calculations from the Polynomial package for that. There seems to be a correspondence between Julia types and structures and the algebraic stuctures I want to work with. So the idea was born to define abstract and concrete types in Julia, the objects of those types representing the ring elements to operate on. As types are first class objects in Julia it was also possible to define combinations in a language affine way. Also ring homomorphisms, i.e. strucure-respecting mappings between rings (of differnt kind) find a natural representation as one-argument-functions or methods with corresponding domains. The typical canonical homomorphisms, can be conveniently implemented as constructors.

The exploitation of the julia structures is in contrast to the alternative package AbstractAlgebra, which defines separate types for ring elements and the ring classes themselves, the elements keeping an explicit link to the owner structure.

To distinct variants of rings, we use type parameters, for example the m in ℤ/m or the x in ℤ[:x]. Other type parameters may be used to specify implementation restictions, for example typically the integer types used for the representation of the objects.

Correspondence between algebraic and Julia categories:

algebraic Julia example
category Ring abstract type abstract type Ring ...
algebraic structure ℤ/m concrete type struct ZZmod{m,Int} <: Ring
specialisation is a Ring type inclusion ZZ{Int] <: Ring
ring element a of R object a isa R
basic binary operations a + b binary operator a + b
homomorphism h : R -> S method h(::R)::S = ...
canonical h : R -> S constructor S(::R) = ...

Usage example

julia> using CommutativeRings

 # starting with some calculation in the quotient field Z/31

julia> m = 31
julia> ZZp = Int8/m
julia> modulus(ZZp)
julia> z1 = ZZp(12)
julia> z2 = ZZp(17)
julia> z1 + z2
julia> z1 * z2
julia> inv(z1)
julia> 13z1

 # using a big prime as parameter, the class is identified by an arbitrary symbol (:p)

julia> ZZbig = BigInt / (big(2)^521 - 1)

julia> modulus(ZZbig)
julia> zb = ZZbig(10)
julia> zb^(modulus(ZZbig)-1) # Fermat's little theorem for primes

... # now polynomials with element type of Z/31

julia> P = UnivariatePolynomial{:x,ZZp}
julia> x = P([0, 1])

julia> p = (x + 2)^2 * (x-1)
x^3 + 3°x^2 - 4°
julia> p.coeff
4-element Array{ZZmod{31,Int8},1}:
julia> 1 + p
x^3 + 3°x^2 - 3°

julia> gcd(p, x-1)
x - 1°
julia> p / (x-1)
x^2 + 4°x + 4°
julia> p / (x+2)
x^2 + x - 2°

julia> p / (x + 3) * (x + 1)
ERROR: DomainError with (x^3 + 3°x^2 - 4°, x + 3°):
not dividable a/b.

Installation of this WIP version

    $ cd ~/.julia/dev

    $ git clone CommutativeRings

    $ julia

    # press "]"
    (v1.3) pkg> activate CommutativeRings
    Activating environment at `~/.julia/dev/CommutativeRings/Project.toml`

    (CommutativeRings) pkg>

     # press backspace

julia> using CommutativeRings
[ Info: Precompiling CommutativeRings [a6d4fa9c-9e0b-4795-89f3-f481b7b5e384]

(CommutativeRings) pkg> test
    Testing CommutativeRings

Test Summary: | Pass  Total
generic       |   23     23
Test Summary: | Pass  Total
typevars      |    8      8
Test Summary: | Pass  Total
ZZ            |  113    113
Test Summary: | Pass  Total
QQ            |   42     42
Test Summary: | Pass  Total
ZZmod         |  271    271
Test Summary: | Pass  Total
univarpolynom |  256    256
Test Summary:   | Pass  Total
multivarpolynom |  151    151
Test Summary: | Pass  Total
ideal         |   21     21
Test Summary: | Pass  Total
fraction      |   32     32
Test Summary: | Pass  Total
quotient      |   24     24
Test Summary: | Pass  Total
factorization |   20     20
Test Summary: | Pass  Total
galoisfields  |   99     99
Test Summary: | Pass  Total
numbertheory  |   24     24
Test Summary: | Pass  Total
enumerations  |   31     31
Test Summary: | Pass  Total
linearalgebra |   23     23
    Testing CommutativeRings tests passed  

Implementation details


Name supertype description / constructor remarks
Ring Any abstract - supertype of all ring classes
FractionField Ring abstract - ring of fractions over a ring
QuotientRing Ring abstract - quotient (or factor-) ring of a ring
Polynomial Ring abstract - polynomials over a ring
ZZ{type} Ring integer numbers type is an integer Julia type
ZZmod{M,type} QuotientRing ZZ / m quotient class modulo m m is an integer Julia value of type M
QQ{type} FractionField rational numbers over integer like Rational{type} - supports integer Julia and integer Rings
Frac{R} FractionField fractions over a R typically polynomials
Quotient{m,R} QuotientRing also R/m, ring modulo m m is an element or an ideal of R
UnivariatePolynomial{X,R} Polynomial also R[:x], ring of polynomials over R X is a symbol like :x
MultivariatePolynomial{X,R} Polynomial also R[:x,:y,...] X is a list of distinct variable names
GaloisField QuotientRing GF(p^r) - efficient implementation of Galois fields more details see below

class construction

Each complete Julia type (with all type parameters specified) defines a singlton algebraic class. Sometimes it is necessary to use distinguishing symbols as a first type parameter if the parameter value cannot be used directly. For that purpose, there is a special function new_class:

    m = big"....."
    Zm = new_class(ZZmod{:p,BigInt}, m)

 # as opposed to

    p = Int128(2)^127 - 1
    Zp = ZZmod{p,Int128}

For general quotient classes and for polynomials there are convenient constructors, which look like the typical mathematical notation R[:x,:y,...] and R/I. Here the symbols :x, :y define the name of the variables of a uni- or multivariate polynomial ring over R. I is an ideal of R or an element of R, which represents the corresponding principal ideal.

    S = ZZ{Int}
    P = S[:x]
    x = monom(P, 1) # same as P([0,1])
    Q = P/(x^2 + 1)

The / notion is also implemented for Julia integer types, so this works:

    Z31 = Int8/31    # equivalent to ZZmod{31,Int8}
    Zbig = BigInt/(big"2"^521-1) # equivalent to new_class(ZZmod{gensym(),BigInt}, m)


Ideals are can be denoted as Ideal([a, ...]) where a, ... are elements of R or Ideal(a). For convenience, principal ideals support also a * R == Ideal(a) and p*R == Ideal(R(p)), where p is an integer.

    Z = ZZ{Int8}
    Z1 = Z/31
    Z2 = Z/Z(31)
    Z3 = Z/31Z
    Z4 = Z/Ideal(Z(31))

    Z1 == Z2 == Z3 == Z4

It may be noted, that 0R is the zero ideal (containing only the zero element of R) and u*R == R for all unit elements u of R. We also have R/0R == R and R/1R == 0R.

If there is one generating element of an ideal, a*R, internally a unit multiple of a is stored to achieve a standard form, for example a monic univariate polynomial.

In the case of multiple generating elements, a, b..., an attempt is made to standardize and reduce the stored base. For example if R is an integral domain, then gcd(a, b...) is stored.

Ideals are most useful with multivariate polynomials, when they are best represented by a minimal base - see below.

Constructors for elements

The class names of all concrete types serve also as constructor names. That means, if R is a class name, then R(a) is an element of R for all a, which are integers or elements of (other) rings, which can be natuarally embedded into R.

For polynomial rings P, the method call monom(P, i) constructs the monic monomials x^i for non-negative integers i. That is extended to multivariate cases monom(P, i, j, ...).

Mathematical operations

operation operator remarks
add + operations with Base.BitIntegers throw upon overflow
subtract - also unary
multiply *
integer power ^ use Base.power_by_squares
divide / only if dividable without remainder
divrem complete integer division
div ÷ quotient integer division
rem % remainder integer division
gcd classical Euclid's algorithm
gcdx extended Euclid's algorithm
pdivrem pseudo division for polynomials over rings d, r = divrem(p, q) => q * d + r = f * p wheref is in the base ring
pgcd pseudo gcd g, f = pgcd(p, q)
pgcdx pseudo gcdx g, u, v, f = pgcdx(p, q) => p * u + q * v = g * f where f is in base ring
iszero test if element is zero-element of its ring
zero zero element of ring
isone test if element is one-element of its ring
one one element of ring
isunit test if element is invertible in its ring
deg degree of polynomial, -1 for zero. For non-polynomials always 0.
lc leading coefficient of polynomial, otherwise identity
ismonomial short for isone(lc(x))
ismonic polynomials of the form c * x^k for c in the base ring, k >= 0 integer
monom return monomial polygon with given degree
isirreducible polynomial cannot be split into non-trivial factors
irreducibles generate all irreducible polynomials of given degree
modulus for quotient rings and Galois fields the defining polynomial
characteristic of ring: smallest positive integer c with c * one(G) == 0, otherwise 0
dimension of Galois fields or vector spaces
order of ring: number of all elements of ring; 0 if infinite
order of element x: smallest positive integer c with x^c == 1, otherwise 0
basetype of ring: type of representative. If no a nested type, the type itself
depth of ring: nesting depth
value representant of element. For R/I the stored value from R. For Galois fields the polynomial.

Associated classes

Each algebraic structure corresponds to a parameterized Julia type or struct. For example, to represent Z/m, there is

    abstract type Ring{<:RingClass} end

    struct ZZmod{m,T<:Integer} <: Ring{ZZmodRingClass}; val::T; end

The subtypes of RingClass are currently only containers for constant type variables. It may be necessary to hold values, which are specific for the algebraic structure, and cannot be stored in as type paramters. That is for example the case, when the modulus m in the example above is a BigInt. In other cases, the classes are unused. The user needs not deal with those types. Access to the type variable is used within the implementation by method owner(::Type{<:Ring}} which provides the RingClass object, when the complete type is known. Preferred operation mode is to take the type parameters directly.

Univariate Polynomials

For each ring type R the class of polynomials over R is created like P = R[:x] where the symbol :x defines the variable name.

Polynomials of this class are obtained by the constructor g = UnivariatePolynomial{R,:x}([1, 2, 3]) or more conveniently by

x = monom(P)
g = 3x^2 + 2x + 1  

If R is a finite Field (that means ZZ/p or GaloisField - see below) the following options are available:

Univariate polynomials may be checked by isirreducible(p) for their irreducibility and factor(g) delivers the list of irreducible factors of g. The factorization is also implemented for univariate polynomials over the integers (for example of type UnivariatePolynomials{ZZ{BigInt}[:x]})

The function irreducible(P, r) delivers the first irreducible polynomial with degree r. All irreducible polynomials of P with degree r are obtained by irreducibles(P, r) which is an iterator. That allows to apply Iterators.filter or find on this list.

While the number of polynomials of degree r is order(R)^r, the subset of irreducibles has order num_irreducibles(P, r).

Galois Fields

All finite field have order p^r where p is a prime number and r >= 1 an integer. It can be represented as a quotient ring of univariate polynomials over ZZ/p by an irreducible monic polynomial g of degree r. In short, if g is known, we have (ZZ/p)[:x] / g as a working implementation of a Galois field. For r == 1 this can be identified with ZZ/p (using g(x) = x).

g can be any monic polynomial of degree r. When constructing the class GF(p^r) = GaloisField{p,r} a brute force search for such polynomials is performed using an efficient method to detect irreducibility. For r > 1 the monom x ∈ (ZZ/p)[:x] / g with 0 and 1 generates the field by applying addition and multiplication operations. Calculated in GF, we have always g(x) = 0. We restrict the selection of g in order to x let generate the multiplicative subgroup of GF by multiplication. That is possible for all p, r.

Time efficiency of algebraic operations is improved by avoiding the expensive multiplicative calculations in the quotient ring and the use of logarithmic tables in the size of p^r. Each element is represented by an integer in 0:p^r-1, which corresponds to a polynomial of degree < r in a canonical manner (for example the number 2p^3 + p + 1 maps uniquley to 2x^3 + x + 1).

Using Galois Fields

We construct a Galois field conveniently by GF(p^r).

julia> p, r = 5, 6; # `p` is prime number, `r` not too big

julia> G = GF(5^6) # GF(5, 6) is also possible

julia> g = modulus(G) # the selected irreducible polynomial
γ^6 + γ + 2°

julia> order(G) # `p^r`

julia> x = G(p) # an easy way to obtain the standard monom

julia> order(x) # `x` generates the multiplicative subgroup

julia> G.(0:p) # element number `p` is `x`
6-element Array{GaloisField{5,6},1}:

julia> g(x) # the monom `x` is a root of `g`

# These are all zeros of g - see also `allzeros`
julia> x.^p.^(0:r-1)
6-element Array{GaloisField{5,6},1}:

julia> g.(x.^p.^(0:r-1)) |> unique
1-element Array{GaloisField{5,6},1}:

Linear Algebra

Matrices and vectors of ring elements are supported. x - A is understood as x * I - A.

The following methods handle vector spaces und subspaces:

operation remarks
nullspace null space (kernel) of matrix
intersect instesection of subspaces
sum sum of subspaces
rank rank of matrix

For a square matrix, also the following methods exist:

operation remarks
inv matrix inverse - using generic LU factorization
det determinant - using generic LU factorization
adjugate for regular A: inv(A) * det(A)
characteristic_polynomial p(A) == 0
companion collides with Polynomials.companion

For inv, det, and adjugate if the element type is P<:Polynomial, it should be widened to Frac(P).

Multivariate Polynomials

Some example usage:

julia> Z = ZZ / 7

julia> P = Z[:x,:y]

julia> x, y = generators(P);

julia> (x + y)^2
x^2 + 2°*x*y + y^2

julia> (x + y)^7
x^7 + y^7

julia> z = (x + y^2) * (x^2 + y)
x^2*y^2 + x^3 + y^3 + x*y

julia> z^2 / (x^2 + y)^2
y^4 + 2°*x*y^2 + x^2

julia> I = [x+2; (x+1)*y];
2-element Array{MultivariatePolynomial{ZZmod{7,Int8},2,Symbol("8693009651133194268"),Int64,Tuple{2}},1}:
 x + 2°
 x*y + y

julia> groebnerbase(I)
2-element Array{MultivariatePolynomial{ZZmod{7,Int8},2,Symbol("8693009651133194268"),Int64,Tuple{2}},1}:
 x + 2°

 julia> P = (ZZ/97)[:x, :y]

julia> x, y = generators(P);

julia> I = [x^2*y + x*y; x*y^2 + 1]
2-element Array{MultivariatePolynomial{ZZmod{97,Int8},2,Symbol("8693009651133194268"),Int64,Tuple{2}},1}:
 x^2*y + x*y
 x*y^2 + 1°

julia> groebnerbase(I)
2-element Array{MultivariatePolynomial{ZZmod{97,Int8},2,Symbol("8693009651133194268"),Int64,Tuple{2}},1}:
 y^2 + 96°
 x + 1°

# example from [Gröbner Base](
julia> I = [x^2 - y; x^3 - x]
2-element Array{MultivariatePolynomial{ZZmod{7,Int8},2,Symbol("8693009651133194268"),Int64,Tuple{2}},1}:
 x^2 - y
 x^3 - x

julia> groebnerbase(I)
3-element Array{MultivariatePolynomial{ZZmod{7,Int8},2,Symbol("8693009651133194268"),Int64,Tuple{2}},1}:
 x^2 - y
 x*y - x
 y^2 - y


This package was inspired by the C++ library CoCoALib, which can be found here: CoCoALib.

The factorization of integer polynomials follows the D. Knuths infamous book "The Art of Computer Programming" chapter 4.6.2.

Copyright © 2019-2021 by Klaus Crusius. This work is released under The MIT License

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