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Introduction

What is Algebraeon?

Algebraeon is a computer algebra system written purely in Rust. It implements algorithms for working with matrices, polynomials, algebraic numbers, factorizations, etc. The focus is on exact algebraic computations over approximate numerical solutions.

Source Code and Contributing

The project is open source and is hosted on GitHub here. Contributions are welcome and should be made via GitHub.

Stability

The API is subject to large breaking changes at this time. I hope to stabilize things more in the not too distant future.

Crates

Algebraeon is published to crates.io under an umbrella crate algebraeon and re-exports the sub-crates:

Algorithms

To give a flavour of what Algebraeon can do some of the algorithms it implements are listed:

  • Euclids algorithm for GCD and the extended version for obtaining Bezout coefficients.
  • Polynomial GCD computations using subresultant pseudo-remainder sequences.
  • AKS algorithm for natural number primality testing.
  • Matrix algorithms including:
    • Putting a matrix into Hermite normal form. In particular putting it into echelon form.
    • Putting a matrix into Smith normal form.
    • Gram–Schmidt algorithm for orthogonalization and orthonormalization.
    • Putting a matrix into Jordan normal.
    • Finding the general solution to a linear or affine system of equations.
  • Polynomial factoring algorithms including:
    • Kronecker's method for factoring polynomials over the integers (slow).
    • Berlekamp-Zassenhaus algorithm for factoring polynomials over the integers.
    • Berlekamp's algorithm for factoring polynomials over finite fields.
    • Cantor–Zassenhaus algorithm for factoring polynomials over finite fields.
    • Trager's algorithm for factoring polynomials over algebraic number fields.
  • Expressing symmetric polynomials in terms of elementary symmetric polynomials.
  • Computations with algebraic numbers:
    • Real root isolation and arithmetic.
    • Complex root isolation and arithmetic.
  • Computations with multiplication tables for small finite groups.
  • Todd-Coxeter algorithm for the enumeration of finite index cosets of a finitely generated groups.

Getting Started

As a Library

Run cargo add algebraeon in the root of your rust project to add the latest version of Algebraeon to your dependencies in your Cargo.toml

[dependencies]
algebraeon = "0.0.11"

Copy one of the example in the next section to get started.

Quickstart Examples

Factoring Integers

To factor large integers using Algebraeon

use algebraeon::sets::structure::ToStringSignature;
use algebraeon::{nzq::Natural, rings::natural::factorization::factor};
use algebraeon::{
    rings::natural::factorization::NaturalCanonicalFactorizationStructure,
    sets::structure::MetaType,
};
use std::str::FromStr;

let n = Natural::from_str("706000565581575429997696139445280900").unwrap();
let f = factor(n.clone()).unwrap();
println!(
    "{} = {}",
    n,
    Natural::structure().factorizations().to_string(&f)
);;
/*
Output:
    706000565581575429997696139445280900 = 2^2 × 5^2 × 6988699669998001 × 1010203040506070809
*/

Algebraeon implements Lenstra elliptic-curve factorization for quickly finding prime factors with around 20 digits.

Factoring Polynomials

Factor the polynomials \(x^2 - 5x + 6\) and \(x^{15} - 1\).

use algebraeon::rings::{polynomial::*, structure::*};
use algebraeon::nzq::Integer;

let x = &Polynomial::<Integer>::var().into_ergonomic();
let f = (x.pow(2) - 5*x + 6).into_verbose();
println!("f(λ) = {}", f.factor().unwrap());
/*
Output:
    f(λ) = 1 * ((-2)+λ) * ((-3)+λ)
*/

let f = (x.pow(15) - 1).into_verbose();
println!("f(λ) = {}", f.factor().unwrap());
/*
Output:
    f(λ) = 1 * ((-1)+λ) * (1+λ+λ^2) * (1+λ+λ^2+λ^3+λ^4) * (1+(-1)λ+λ^3+(-1)λ^4+λ^5+(-1)λ^7+λ^8)
*/

so

\[x^2 - 5x + 6 = (x-2)(x-3)\]

\[x^{15}-1 = (x-1)(x^2+x+1)(x^4+x^3+x^2+x+1)(x^8-x^7+x^5-x^4+x^3-x+1)\]

Linear Systems of Equations

Find the general solution to the linear system

\[a \begin{pmatrix}3 \\ 4 \\ 1\end{pmatrix} + b \begin{pmatrix}2 \\ 1 \\ 2\end{pmatrix} + c \begin{pmatrix}1 \\ 3 \\ -1\end{pmatrix} = \begin{pmatrix}5 \\ 5 \\ 3\end{pmatrix}\]

for integers \(a\), \(b\) and \(c\).

use algebraeon::nzq::Integer;
use algebraeon::rings::module::finitely_free_module::RingToFinitelyFreeModuleSignature;
use algebraeon::rings::matrix::Matrix;
use algebraeon::sets::structure::MetaType;
let m = Matrix::<Integer>::from_rows(vec![vec![3, 4, 1], vec![2, 1, 2], vec![1, 3, -1]]);
let y = vec![5.into(), 5.into(), 3.into()];
for x in Integer::structure()
    .free_module(3)
    .affine_subsets()
    .affine_basis(&m.row_solution_set(&y))
{
    println!("{:?}", x);
}
/*
Output:
    [Integer(0), Integer(2), Integer(1)]
    [Integer(1), Integer(1), Integer(0)]
*/

so two solutions are given by \((a, b, c) = (0, 2, 1)\) and \((a, b, c) = (1, 1, 0)\) and every solution is a linear combination of these two solutions; The general solution is given by all \((a, b, c)\) such that

\[\begin{pmatrix}a \\ b \\ c\end{pmatrix} = s\begin{pmatrix}0 \\ 2 \\ 1\end{pmatrix} + t\begin{pmatrix}1 \\ 1 \\ 0\end{pmatrix}\]

where \(s\) and \(t\) are integers such that \(s + t = 1\).

Complex Root Isolation

Find all complex roots of the polynomial \[f(x) = x^5 + x^2 - x + 1\]

use algebraeon::rings::{polynomial::*, structure::*};
use algebraeon::nzq::Integer;

let x = &Polynomial::<Integer>::var().into_ergonomic();
let f = (x.pow(5) + x.pow(2) - x + 1).into_verbose();
// Find the complex roots of f(x)
for root in f.all_complex_roots() {
    println!("root {} of degree {}", root, root.degree());
}
/*
Output:
    root ≈-1.328 of degree 3
    root ≈0.662-0.559i of degree 3
    root ≈0.662+0.559i of degree 3
    root -i of degree 2
    root i of degree 2
*/

Despite the output, the roots found are not numerical approximations. Rather, they are stored internally as exact algebraic numbers by using isolating boxes in the complex plane.

Factoring Multivariable Polynomials

Factor the following multivariable polynomial with integer coefficients

\[f(x, y) = 6x^4 - 6x^3y^2 + 6xy - 6x - 6y^3 + 6y^2\]

use algebraeon::{nzq::Integer, rings::{polynomial::*, structure::*}};

let x = &MultiPolynomial::<Integer>::var(Variable::new("x")).into_ergonomic();
let y = &MultiPolynomial::<Integer>::var(Variable::new("y")).into_ergonomic();

let f = (6 * (x.pow(4) - x.pow(3) * y.pow(2) + x * y - x - y.pow(3) + y.pow(2))).into_verbose();
println!("f(x, y) = {}", f.factor().unwrap());

/*
Output:
    f(x, y) = 1 * ((3)1) * ((2)1) * (x+(-1)y^2) * (x^3+y+(-1)1)
*/

so the factorization of \(f(x, y)\) is

\[f(x, y) = 2 \times 3 \times (x^3 + y - 1) \times (y^2 - x)\]

P-adic Root Finding

Find the \(2\)-adic square roots of \(17\).

use algebraeon::nzq::{Natural, Integer};
use algebraeon::rings::{polynomial::*, structure::*};
let x = Polynomial::<Integer>::var().into_ergonomic();
let f = (x.pow(2) - 17).into_verbose();
for mut root in f.all_padic_roots(&Natural::from(2u32)) {
    println!("{}", root.truncate(&20.into()).string_repr()); // Show 20 2-adic digits
}
/*
Output:
    ...00110010011011101001
    ...11001101100100010111
*/

Truncating to the last 16 bits it can be verified that, modulo \(2^{16}\), the square of these values is \(17\).

let a = 0b0010011011101001u16;
assert_eq!(a.wrapping_mul(a), 17u16);
let b = 0b1101100100010111u16;
assert_eq!(b.wrapping_mul(b), 17u16);

Enumerating a Finitely Generated Group

Let \(G\) be the finitely generated group generated by \(3\) generators \(a\), \(b\), \(c\) subject to the relations \(a^2 = b^2 = c^2 = (ab)^3 = (bc)^5 = (ac)^2 = e\).

\[G = \langle a, b, c : a^2 = b^2 = c^2 = (ab)^3 = (bc)^5 = (ac)^2 = e \rangle\]

Using Algebraeon, \(G\) is found to be a finite group of order \(120\):

use algebraeon::groups::free_group::todd_coxeter::*;
let mut g = FinitelyGeneratedGroupPresentation::new();
// Add the 3 generators
let a = g.add_generator();
let b = g.add_generator();
let c = g.add_generator();
// Add the relations
g.add_relation(a.pow(2));
g.add_relation(b.pow(2));
g.add_relation(c.pow(2));
g.add_relation((&a * &b).pow(3));
g.add_relation((&b * &c).pow(5));
g.add_relation((&a * &c).pow(2));
// Count elements
let (n, _) = g.enumerate_elements();
assert_eq!(n, 120);

Jordan Normal Form of a Matrix

use algebraeon::nzq::{Rational};
use algebraeon::rings::{matrix::*, isolated_algebraic::*};
use algebraeon::sets::structure::*;
// Construct a matrix
let a = Matrix::<Rational>::from_rows(vec![
    vec![5, 4, 2, 1],
    vec![0, 1, -1, -1],
    vec![-1, -1, 3, 0],
    vec![1, 1, -1, 2],
]);
// Put it into Jordan Normal Form
let j = MatrixStructure::new(ComplexAlgebraic::structure()).jordan_normal_form(&a);
j.pprint();
/*
Output:
    / 2    0    0    0 \
    | 0    1    0    0 |
    | 0    0    4    1 |
    \ 0    0    0    4 /
*/

Computing Discriminants

Algebraeon can find an expression for the discriminant of a polynomial in terms of the polynomials coefficients.

use algebraeon::rings::polynomial::*;
use algebraeon::nzq::Integer;

let a_var = Variable::new("a");
let b_var = Variable::new("b");
let c_var = Variable::new("c");
let d_var = Variable::new("d");
let e_var = Variable::new("e");

let a = MultiPolynomial::<Integer>::var(a_var);
let b = MultiPolynomial::<Integer>::var(b_var);
let c = MultiPolynomial::<Integer>::var(c_var);
let d = MultiPolynomial::<Integer>::var(d_var);
let e = MultiPolynomial::<Integer>::var(e_var);

let p =
    Polynomial::<MultiPolynomial<Integer>>::from_coeffs(vec![c.clone(), b.clone(), a.clone()]);
println!("p(λ) = {}", p);
println!("disc(p) = {}", p.discriminant().unwrap());

println!();

let p = Polynomial::<MultiPolynomial<Integer>>::from_coeffs(vec![
    d.clone(),
    c.clone(),
    b.clone(),
    a.clone(),
]);
println!("p(λ) = {}", p);
println!("disc(p) = {}", p.discriminant().unwrap());

println!();

let p = Polynomial::<MultiPolynomial<Integer>>::from_coeffs(vec![
    e.clone(),
    d.clone(),
    c.clone(),
    b.clone(),
    a.clone(),
]);
println!("p(λ) = {}", p);
println!("disc(p) = {}", p.discriminant().unwrap());

/*
Output:
    p(λ) = (c)+(b)λ+(a)λ^2
    disc(p) = (-4)ac+b^2

    p(λ) = (d)+(c)λ+(b)λ^2+(a)λ^3
    disc(p) = (-27)a^2d^2+(18)abcd+(-4)ac^3+(-4)b^3d+b^2c^2
*/

so

\[\mathop{\text{disc}}(ax^2 + bx + c) = b^2 - 4ac\]

\[\mathop{\text{disc}}(ax^3 + bx^2 + cx + d) = b^2c^2 - 4ac^3 - 4b^3d - 27a^2d^2 + 18abcd\]

The Structure System

This section describes how Algebraeon represents the mathematical idea of sets with some additional structure such as groups, rings, fields, and so on.

Structure Types

Motivation

In mathematics there are many instances of sets with some additional structure, for example:

  • The set of integers with its ordered ring structure.
  • The set of rational numbers with its ordered field structure.
  • For each natural number \(n \in \mathbb{N}\), the finite set of integers modulo \(n\) with its ring structure.
  • The set of all algebraic numbers in \(\mathbb{C}\) with its field structure.
  • The set of all ideals in a ring with the operations of ideal addition, ideal intersection, and ideal multiplication. Depending on the ring, ideals may be uniquely factorable as a product of prime ideals.

The approach taken by Algebraeon to represent such sets with additional structure is well illustrated by the example of the ring of integers modulo \(n \in \mathbb{N}\). This would be done as follows:

  • Define a structure type IntegersModuloN whose objects shall represent the ring of integers modulo \(n\) for different values of \(n\).
  • Implement the desired structure by implementing signature traits on the structure type. In the case of IntegersModuloN the required signature traits are:
    • Signature the base signature trait.
    • SetSignature<Set = Integer> so that instances of Integer shall be used to represent elements of the set of integers modulo \(n\).
    • EqSignature so that pairs of Integers can be tested for equality modulo \(n\).
    • FiniteSetSignature so that a list of all integers modulo \(n\) can be produced.
    • SemiRingSignature so that Integers can be added and multiplied modulo \(n\).
    • RingSignature so that Integers can be subtracted modulo \(n\).

In practice, this could look like

use algebraeon::nzq::{Integer, Natural};
use algebraeon::rings::structure::*;
use algebraeon::sets::structure::*;

#[derive(Debug, Clone, PartialEq, Eq)]
struct IntegersModuloN {
    n: Natural,
}

impl Signature for IntegersModuloN {}

impl SetSignature for IntegersModuloN {
    type Set = Integer;

    fn is_element(&self, x: &Integer) -> Result<(), String> {
        if x >= &self.n {
            return Err("too big".to_string());
        }
        Ok(())
    }
}

impl EqSignature for IntegersModuloN {
    fn equal(&self, a: &Integer, b: &Integer) -> bool {
        a == b
    }
}

use algebraeon::rings::structure::{RingSignature, SemiRingSignature};

impl AdditiveMonoidSignature for IntegersModuloN {
    fn zero(&self) -> Self::Set {
        Integer::ZERO
    }

    fn add(&self, a: &Self::Set, b: &Self::Set) -> Self::Set {
        ((a + b) % &self.n).into()
    }
}

impl AdditiveGroupSignature for IntegersModuloN {
    fn neg(&self, a: &Self::Set) -> Self::Set {
        (-a % &self.n).into()
    }
}

impl SemiRingSignature for IntegersModuloN {
    fn one(&self) -> Self::Set {
        (Integer::ONE % &self.n).into()
    }

    fn mul(&self, a: &Self::Set, b: &Self::Set) -> Self::Set {
        ((a * b) % &self.n).into()
    }
}

impl RingSignature for IntegersModuloN {}

let mod_6 = IntegersModuloN { n: 6u32.into() };
// Since we've given `mod_6` the structure of a ring, Algebraeon implements
// the repeated squaring algorithm for taking very large powers modulo `n`.
assert!(mod_6.equal(
    &mod_6.nat_pow(&2.into(), &1000000000000u64.into()),
    &4.into()
));

Canonical Structures

Sometimes the situation is simple and we only want to define one set with structure rather than a family of sets, for example, the set of all rational numbers. Since sets with structure are represented in Algebraeon objects of structure types we will need a structure type with exactly once instance. This can be done explicitly like so

use algebraeon::{nzq::Rational, rings::structure::*, sets::structure::*};

#[derive(Debug, Clone, PartialEq, Eq)]
pub struct MyRational {
    value: Rational,
}

#[derive(Debug, Clone, PartialEq, Eq)]
pub struct MyRationalCanonicalStructure {}

impl Signature for MyRationalCanonicalStructure {}

impl SetSignature for MyRationalCanonicalStructure {
    type Set = MyRational;

    fn is_element(&self, _x: &Self::Set) -> Result<(), String> {
        Ok(())
    }
}

impl EqSignature for MyRationalCanonicalStructure {
    fn equal(&self, x: &Self::Set, y: &Self::Set) -> bool {
        x == y
    }
}

However, Algebraeon provides a derive macro CanonicalStructure which reduces the boilerplate above to

use algebraeon::{nzq::Rational, rings::structure::*, sets::structure::*};

#[derive(Debug, Clone, PartialEq, Eq, CanonicalStructure)]
pub struct MyRational {
    value: Rational,
}

In any case, once we have the structure type MyRationalCanonicalStructure implementing Signature + SetSignature<Set = MyRational> we can go on to implement more structure traits like RingSignature and FieldSignature for MyRationalCanonicalStructure to give the set of instances of MyRational the structure of a ring or a field.

Natural Numbers

Constructing naturals

There are several ways to construct naturals. Some of them are:

  • Natural::ZERO, Natural::ONE, Natural::TWO
  • from a string: Natural::from_str()
  • from builtin unsigned types Natural::from(4u32)
use std::str::FromStr;
use algebraeon::nzq::Natural;

let one = Natural::ONE;
let two = Natural::TWO;
let n = Natural::from(42usize);
let big = Natural::from_str("706000565581575429997696139445280900").unwrap();

Basic operations

Natural supports the following operators:

  • + (addition)

  • * (multiplication)

  • % (modulo)

For exponentiation, use the method .pow(&exp) instead of ^ (which is xor).

Available functions

use algebraeon::nzq::*;
use algebraeon::rings::natural::factorization::NaturalCanonicalFactorizationStructure;
use algebraeon::sets::structure::*;

let a = Natural::from(12u32);
let b = Natural::from(5u32);

// Basic operations
let sum = &a + &b; // 17
let product = &a * &b; // 60
let power = a.pow(&b); // 248832
let modulo = &a % &b; // 2

assert_eq!(sum, Natural::from(17u32));
assert_eq!(product, Natural::from(60u32));
assert_eq!(power, Natural::from(248832u32));
assert_eq!(modulo, Natural::from(2u32));

// Factorial
assert_eq!(b.factorial(), Natural::from(120u32));

// nth_root_floor and nth_root_ceil
assert_eq!(a.nth_root_floor(&b), Natural::from(1u32));
assert_eq!(a.nth_root_ceil(&b), Natural::from(2u32));

// sqrt_floor, sqrt_ceil, sqrt_if_square
assert_eq!(a.sqrt_floor(), Natural::from(3u32));
assert_eq!(a.sqrt_ceil(), Natural::from(4u32));

let square = Natural::from(144u32); // 12²
assert_eq!(square.sqrt_if_square(), Some(a.clone()));
assert_eq!(a.sqrt_if_square(), None);

// is_square
assert!(square.is_square());
assert!(!a.is_square());

// Combinatorics
assert_eq!(choose(&a, &b), Natural::from(792u32));

// GCD and LCM
assert_eq!(gcd(a.clone(), b.clone()), Natural::from(1u32));
assert_eq!(lcm(a.clone(), b.clone()), Natural::from(60u32));

// is_prime
use algebraeon::rings::natural::factorization::primes::is_prime;
assert!(!is_prime(&a)); // 12 is not prime
assert!(is_prime(&b)); // 5 is prime

// Euler's totient function
use algebraeon::rings::natural::factorization::factor;
assert_eq!(
    Natural::structure()
        .factorizations()
        .euler_totient(&factor(a).unwrap()),
    Natural::from(4u32)
); // φ(12) = 4
assert_eq!(
    Natural::structure()
        .factorizations()
        .euler_totient(&factor(b).unwrap()),
    Natural::from(4u32)
); // φ(5) = 4

Factoring

Algebraeon implements Lenstra elliptic-curve factorization for quickly finding prime factors up to around 20 digits.

use algebraeon::sets::structure::ToStringSignature;
use algebraeon::{nzq::Natural, rings::natural::factorization::factor};
use algebraeon::{
    rings::natural::factorization::NaturalCanonicalFactorizationStructure,
    sets::structure::MetaType,
};
use std::str::FromStr;

let n = Natural::from_str("706000565581575429997696139445280900").unwrap();
let f = factor(n.clone()).unwrap();
println!(
    "{} = {}",
    n,
    Natural::structure().factorizations().to_string(&f)
);;
/*
Output:
    706000565581575429997696139445280900 = 2^2 × 5^2 × 6988699669998001 × 1010203040506070809
*/

Integers

Constructing integers

There are several ways to construct integers. Some of them are:

  • Integer::ZERO, Integer::ONE, Integer::TWO
  • from a string: Integer::from_str()
  • from builtin unsigned or signed types Integer::from(-4i32)
  • from a Natural: Integer::from(Natural::TWO)
use std::str::FromStr;
use algebraeon::nzq::{Integer, Natural};

let one = Integer::ONE;
let two = Integer::TWO;
let n = Integer::from(42usize);
let m = Integer::from(-42);
let big = Integer::from_str("706000565581575429997696139445280900").unwrap();
let from_natural = Integer::from(Natural::from(5u32));

Basic operations

Integer supports the following operators:

  • + (addition)

  • - (subtraction or negation)

  • * (multiplication)

  • % (modulo)

For exponentiation, use the methods .nat_pow(&exp) or .int_pow(&exp).

Available functions

  • abs
use algebraeon::nzq::{Integer, Natural};
use algebraeon::rings::structure::*;

let a = Integer::from(-12);
let b = Integer::from(5);

// Basic operations
let sum = &a + &b;                              // -7
let neg = -&a;                                  // 12
let sub = &a - &b;                              // -17
let product = &a * &b;                          // -60
let power = a.nat_pow(&Natural::from(5u32));    // -248832
let modulo = &a % &b;                           // 3

assert_eq!(sum, Integer::from(-7));
assert_eq!(neg, Integer::from(12));
assert_eq!(sub, Integer::from(-17));
assert_eq!(product, Integer::from(-60));
assert_eq!(power, Integer::from(-248832));
assert_eq!(modulo, Integer::from(3));

// abs
use algebraeon::nzq::traits::Abs;
let abs_a = a.abs();
assert_eq!(abs_a, Natural::from(12u32));

Rational Numbers

The Rational type represents a number of the form a/b, where a is an integer and b is a non-zero integer. It is a wrapper around malachite_q::Rational.

Constructing rationals

There are several ways to construct rational numbers. Some of them are:

  • Rational::ZERO, Rational::ONE, Rational::TWO, Rational::ONE_HALF
  • from a string: Rational::from_str()
  • from builtin signed/unsigned types: Rational::from(4u32), Rational::from(-3i64)
  • from Integer or Natural: Rational::from(Integer::from(-5))
  • from two integers: Rational::from_integers(n, d)

Basic operations

Rational supports the following operators:

  • + (addition)
  • - (subtraction or negation)
  • * (multiplication)
  • / (division)

Available functions

The following methods are available on Rational:

  • abs() – returns the absolute value
  • floor() – returns the greatest integer less than or equal to the rational
  • ceil() – returns the smallest integer greater than or equal to the rational
  • approximate(max_denominator) – approximates the rational with another having a bounded denominator
  • simplest_rational_in_closed_interval(a, b) – finds the simplest rational between two bounds
  • simplest_rational_in_open_interval(a, b) – finds the simplest rational strictly between two bounds
  • decimal_string_approx() – returns a decimal string approximation of the rational
  • exhaustive_rationals() – returns an infinite iterator over all reduced rational numbers
  • into_abs_numerator_and_denominator() – returns the absolute numerator and denominator as Naturals
  • try_from_float_simplest(x: f64) – converts a float into the simplest rational representation
use std::str::FromStr;
use algebraeon::nzq::{Rational, Integer, Natural};

let zero = Rational::ZERO;
let one = Rational::ONE;
let half = Rational::ONE_HALF;

let r1 = Rational::from(5u32);
let r2 = Rational::from(-3i64);
let r3 = Rational::from(Integer::from(-7));
let r4 = Rational::from(Natural::from(10u32));
let r5 = Rational::from_str("42/7").unwrap();
let r6 = Rational::from_integers(3, 4);

let a = Rational::from_str("2/3").unwrap();
let b = Rational::from_str("1/6").unwrap();

// Basic operations
let sum = &a + &b;        // 5/6
let diff = &a - &b;       // 1/2
let product = &a * &b;    // 1/9
let quotient = &a / &b;   // 4
let negated = -&a;        // -2/3

// Available functions
let r = Rational::from_str("-2/5").unwrap();

// abs
use algebraeon::nzq::traits::Abs;
assert_eq!(r.clone().abs(), Rational::from_str("2/5").unwrap());

// ceil
use algebraeon::nzq::traits::Ceil;
assert_eq!(r.clone().ceil(), Integer::from(0));

// floor
use algebraeon::nzq::traits::Floor;
assert_eq!(r.clone().floor(), Integer::from(-1));

// into_abs_numerator_and_denominator
use algebraeon::nzq::traits::Fraction;
let (n, d) = r.into_abs_numerator_and_denominator();
assert_eq!(n, Natural::from(2u32));
assert_eq!(d, Natural::from(5u32));

// approximate
let approx = Rational::from_str("355/113").unwrap()
    .approximate(&Natural::from(10u32));
assert_eq!(approx, Rational::from_str("22/7").unwrap());

// simplest_rational_in_closed_interval
let a = Rational::from_str("2/5").unwrap();
let b = Rational::from_str("3/5").unwrap();
let simple = Rational::simplest_rational_in_closed_interval(&a, &b);
assert_eq!(simple, Rational::from_str("1/2").unwrap());

// simplest_rational_in_open_interval
let open_a = Rational::from_str("1/3").unwrap();
let open_b = Rational::from_str("2/3").unwrap();
let open_simple = Rational::simplest_rational_in_open_interval(&open_a, &open_b);
assert!(open_simple > open_a && open_simple < open_b);

// try_from_float_simplest
let from_float = Rational::try_from_float_simplest(0.5).unwrap();
assert_eq!(from_float, Rational::ONE_HALF);

// decimal_string_approx
let dec = simple.decimal_string_approx();
assert_eq!(dec, "0.5");

// Iteration
let mut iter = Rational::exhaustive_rationals();
assert_eq!(iter.next(), Some(Rational::ZERO));
assert_eq!(iter.next(), Some(Rational::ONE));

Free Modules

This section explains how to work with free modules, submodules and cosets of submodules.

If \(R\) is a commutative ring then the set \(R^n\) of all length \(n\) tuples of elements from \(R\) is a free \(R\)-module with standard basis \(e_1, \dots, e_n \in R^n\). \[e_1 = (1, 0, \dots, 0)\] \[e_2 = (0, 1, \dots, 0)\] \[\vdots\] \[e_n = (0, 0, \dots, 1)\]

Algebraeon supports working with free modules over the following rings \(R\):

  • The integers \(\mathbb{Z}\)
  • The rationals \(\mathbb{Q}\)
  • Any field
  • Any Euclidean domain

The examples in this section primarily illustrate how to use Algebraeon in the case \(R = \mathbb{Z}\).

Free Modules

The free module \(R^n\) is represented by objects of type FinitelyFreeModuleStructure. A free module structure can be obtained from the ring of scalars by calling .free_module(n) (the module will take the scalar ring structure by reference) or .into_free_module(n) (the module will take ownership of the scalar ring structure).

For example, to obtain \(\mathbb{Z}^3\)

use algebraeon::nzq::Integer;
use algebraeon::rings::module::finitely_free_module::RingToFinitelyFreeModuleSignature;
use algebraeon::sets::structure::MetaType;

let module = Integer::structure().into_free_module(3);

Elements of \(\mathbb{Z}^3\) are represented by objects of type Vec<Integer> and basic operations with the elements are provided by the module structure.

use algebraeon::nzq::Integer;
use algebraeon::rings::module::finitely_free_module::*;
use algebraeon::rings::structure::*;
use algebraeon::sets::structure::*;

let module = Integer::structure().into_free_module(3);

let a = vec![1.into(), 2.into(), 3.into()];
let b = vec![(-1).into(), 2.into(), (-2).into()];

assert!(module.equal(
    &module.neg(&a),
    &vec![(-1).into(), (-2).into(), (-3).into()]
));

assert!(
    module.equal(&module.add(&a, &b), 
    &vec![0.into(), 4.into(), 1.into()]
));

assert!(
    module.equal(&module.sub(&a, &b),
    &vec![2.into(), 0.into(), 5.into()]
));

assert!(module.equal(
    &module.scalar_mul(&a, &5.into()),
    &vec![5.into(), 10.into(), 15.into()]
));

The scalar ring structure can be obtained from a module by calling .ring().

use algebraeon::nzq::Integer;
use algebraeon::rings::module::finitely_free_module::*;
use algebraeon::sets::structure::*;

let module = Integer::structure().into_free_module(3);

let ring = module.ring();
assert_eq!(ring, &Integer::structure());

Submodules

The set of submodules of the free module \(R^n\) is represented by objects of type FinitelyFreeSubmoduleStructure. This structure can be obtained from a module by calling .submodules() (the structure will take the module structure by reference) or .into_submodules() (the structure will take ownership of the module structure).

For example, to obtain the set of all submodules of \(\mathbb{Z}^3\)

use algebraeon::nzq::Integer;
use algebraeon::rings::module::finitely_free_module::*;
use algebraeon::sets::structure::*;

let submodules = Integer::structure().into_free_module(3).into_submodules();

Submodules of \(R^n\) are represented by objects of type FinitelyFreeSubmodule.

Constructing Submodules

The zero submodule \({0} \subseteq R^n\) can be constructed using .zero_submodule() and the full submodule \(R^n \subseteq R^n\) can be constructed using .full_submodule().

use algebraeon::nzq::Integer;
use algebraeon::rings::module::finitely_free_module::*;
use algebraeon::sets::structure::*;

let submodules = Integer::structure().into_free_module(3).into_submodules();

assert_eq!(submodules.zero_submodule().rank(), 0);
assert_eq!(submodules.full_submodule().rank(), 3);

The submodule given by the span of some elements of the module can be constructed using .span(..).

use algebraeon::nzq::Integer;
use algebraeon::rings::module::finitely_free_module::*;
use algebraeon::sets::structure::*;

let submodules = Integer::structure().into_free_module(3).into_submodules();

assert_eq!(
    submodules
        .span(vec![
            &vec![1.into(), 2.into(), 2.into()],
            &vec![2.into(), 1.into(), 1.into()],
            &vec![3.into(), 3.into(), 3.into()]
        ])
        .rank(),
    2
);

The submodule given by the kernel of some elements can be constructed using .kernel(..).

use algebraeon::nzq::Integer;
use algebraeon::rings::module::finitely_free_module::*;
use algebraeon::sets::structure::*;

let submodules = Integer::structure().into_free_module(3).into_submodules();

assert!(submodules.equal(
    &submodules.kernel(vec![
        &vec![1.into(), 2.into()],
        &vec![2.into(), 1.into()],
        &vec![3.into(), 3.into()],
    ]),
    &submodules.span(vec![&vec![1.into(), 1.into(), (-1).into()]])
));

Basic Operations

Test submodules for equality using .equal(..).

use algebraeon::nzq::Integer;
use algebraeon::rings::module::finitely_free_module::*;
use algebraeon::sets::structure::*;

let submodules = Integer::structure().into_free_module(3).into_submodules();

assert!(submodules.equal(
    &submodules.span(vec![
        &vec![2.into(), 2.into(), 0.into()],
        &vec![2.into(), (-2).into(), 0.into()],
    ]),
    &submodules.span(vec![
        &vec![4.into(), 0.into(), 0.into()],
        &vec![0.into(), 4.into(), 0.into()],
        &vec![2.into(), 2.into(), 0.into()],
    ])
));

assert!(!submodules.equal(
    &submodules.span(vec![
        &vec![1.into(), 1.into(), 0.into()],
        &vec![2.into(), 3.into(), 0.into()],
    ]),
    &submodules.span(vec![
        &vec![1.into(), 2.into(), 3.into()],
        &vec![1.into(), 1.into(), 0.into()],
    ])
));

Check whether a submodule contains an element using .contains_element(..).

use algebraeon::nzq::Integer;
use algebraeon::rings::module::finitely_free_module::*;
use algebraeon::sets::structure::*;

let submodules = Integer::structure().into_free_module(3).into_submodules();

let a = submodules.span(vec![
    &vec![2.into(), 2.into(), 0.into()],
    &vec![2.into(), (-2).into(), 0.into()],
]);

assert!(submodules.contains_element(&a, &vec![4.into(), 4.into(), 0.into()]));
assert!(!submodules.contains_element(&a, &vec![3.into(), 4.into(), 0.into()]));
assert!(!submodules.contains_element(&a, &vec![4.into(), 4.into(), 1.into()]));

Check whether a submodule is a subset of another submodule using .contains(..).

use algebraeon::nzq::Integer;
use algebraeon::rings::module::finitely_free_module::*;
use algebraeon::sets::structure::*;

let submodules = Integer::structure().into_free_module(3).into_submodules();

let a = submodules.span(vec![&vec![3.into(), 3.into(), 3.into()]]);
let b = submodules.span(vec![&vec![6.into(), 6.into(), 6.into()]]);

assert!(submodules.contains(&a, &b));
assert!(!submodules.contains(&b, &a));

Compute the sum of two submodules using .sum(..) and compute the intersection of two submodules using .intersection(..).

use algebraeon::nzq::Integer;
use algebraeon::rings::module::finitely_free_module::*;
use algebraeon::sets::structure::*;

let submodules = Integer::structure().into_free_module(3).into_submodules();

let a = submodules.span(vec![&vec![4.into(), 4.into(), 4.into()]]);
let b = submodules.span(vec![&vec![6.into(), 6.into(), 6.into()]]);

let sum_ab = submodules.span(vec![&vec![2.into(), 2.into(), 2.into()]]);
assert!(submodules.equal(&submodules.sum(&a, &b), &sum_ab));

let intersect_ab = submodules.span(vec![&vec![12.into(), 12.into(), 12.into()]]);
assert!(submodules.equal(&submodules.intersect(&a, &b), &intersect_ab));

Multivariate Polynomials

Example

use algebraeon::nzq::Integer;
use algebraeon::rings::polynomial::*;
use algebraeon::rings::structure::*;
use std::collections::HashMap;

// Define symbols for the variables we wish to use
let x_var = Variable::new("x");
let y_var = Variable::new("y");

// Construct the polynomials x and y from the variables
let x = &MultiPolynomial::<Integer>::var(x_var.clone());
let y = &MultiPolynomial::<Integer>::var(y_var.clone());

// x and y can now be used just like other ring elements in Algebraeon
let f = MultiPolynomial::add(x, y).nat_pow(&3u32.into());

// f = x^3+(3)x^2y+(3)xy^2+y^3
println!("f = {}", f);

// For evaluating f the inputs are specified using the variable symbols
let v = f.evaluate(HashMap::from([(x_var, &1.into()), (y_var, &2.into())]));

// v = f(1, 2) = 27
println!("v = {}", v);

Algebraic Numbers

Let \(F\) and \(K\) be fields with \(F \subseteq K\). A number \(\alpha \in K\) is said to be algebraic over \(F\) if there exists a polynomial \(p(x) \in F[x]\) such that \(f(\alpha) = 0\). If \(\alpha \in K\) is algebraic over \(F\) then the subset \(I\) of \(F[x]\) given by the set of all polynomials \(p(x) \in F[x]\) satisfying \(f(\alpha) = 0\) is an ideal of \(F[x]\). \[I := \{p(x) \in F[x] : p(\alpha) = 0\} \text{ is an ideal of } F[x]\] \(F[x]\) is a principal ideal domain and the unique monic polynomial \(m(x) \in F[x]\) which generates \(I\) is called the minimal polynomial of \(\alpha\).

Algebraon supports the following algebraic numbers:

  • Real numbers \(\alpha \in \mathbb{R}\) which are algebraic over \(\mathbb{Q}\). Represented by the minimal polynomial \(m(x) \in \mathbb{Q}[x]\) of \(\alpha\) and a rational isolating interval \((a, b) \subseteq \mathbb{R}\) consisting a pair of rational numbers \(a, b \in \mathbb{Q}\) such that \(\alpha \in (a, b)\) and no other root of \(m(x)\) is contained in \((a, b)\).
  • Complex numbers \(\alpha \in \mathbb{C}\) which are algebraic over \(\mathbb{Q}\). Represented by the minimal polynomial \(m(x) \in \mathbb{Q}[x]\) of \(\alpha\) and a rational isolating box in the complex plane.
  • For a prime \(p \in \mathbb{N}\), \(p\)-adic numbers \(\alpha \in \mathbb{Q}_p\) which are algebraic over \(\mathbb{Q}\). Represented by the minimal polynomial \(m(x) \in \mathbb{Q}[x]\) of \(\alpha\) and a \(p\)-adic isolating ball consisting of a rational number \(c \in \mathbb{Q}\) and a valuation \(v \in \mathbb{N} \sqcup {\infty}\) such that \(\alpha - c\) has \(p\)-adic valuation at least \(v\) and no other root \(\beta\) of \(m(x)\) is such that \(\beta - c\) has valuation \(p\)-adic at least \(v\).
  • If \(m(x) \in \mathbb{Q}[x]\) is irreducible then the quotient field \(K := \frac{\mathbb{Q}[x]}{m(x)}\) is a finite dimensional extension of \(\mathbb{Q}\), also known as an algebraic number field. Every element of \(K\) is algebraic over \(\mathbb{Q}\). Elements of \(K\) are represented by rational polynomials up to adding rational polynomial multiples of \(m(x)\).

Obtaining Polynomial Roots

This example finds all roots of a rational polynomial \(f(x) \in \mathbb{Q}[x]\) in various extension fields of \(\mathbb{Q}\). The key parts are

  • Calling .into_rational_extension() on a field \(K\) to obtain the field extension \(\mathbb{Q} \to K\).
  • Calling .all_roots(&f) on a field extension \(\mathbb{Q} \to K\) to obtain all roots of \(f(x) \in \mathbb{Q}[x]\) belonging to \(K\).
use algebraeon::nzq::{Natural, Rational};
use algebraeon::rings::isolated_algebraic::ComplexAlgebraic;
use algebraeon::rings::isolated_algebraic::PAdicAlgebraic;
use algebraeon::rings::isolated_algebraic::RealAlgebraic;
use algebraeon::rings::{polynomial::*, structure::*};
use algebraeon::sets::structure::*;

// Find all roots of f in some fields

let f = Polynomial::<Rational>::from_str(
    "(x - 3) * (x^2 - 17) * (x^2 + 1)", "x"
).unwrap();
println!("f = {}", f);

println!();

let two_adic = PAdicAlgebraic::structure(Natural::from(2u32))
    .into_rational_extension();
let three_adic = PAdicAlgebraic::structure(Natural::from(3u32))
    .into_rational_extension();
let complex = ComplexAlgebraic::structure()
    .into_rational_extension();
let real = RealAlgebraic::structure()
    .into_rational_extension();
let anf = Polynomial::from_str("x^2 + 1", "x")
    .unwrap()
    .algebraic_number_field()
    .unwrap()
    .into_rational_extension();

println!("Real roots of f");
for x in real.all_roots(&f) {
    println!("{}", x);
}

println!();

println!("Complex roots of f");
for x in complex.all_roots(&f) {
    println!("{}", x);
}

println!();

println!("2-adic roots of f");
for x in two_adic.all_roots(&f) {
    println!("{}", x);
}

println!();

println!("3-adic roots of f");
for x in three_adic.all_roots(&f) {
    println!("{}", x);
}

println!();

println!("Roots of f in Q[i]");
for x in anf.all_roots(&f) {
    println!(
        "{}",
        x.apply_map_into(MultiPolynomial::constant).evaluate(
            &Rational::structure()
                .into_multivariable_polynomial_ring()
                .var(Variable::new("i"))
        )
    );
}

/*
Output:
    f = (51)+(-17)λ+(48)λ^2+(-16)λ^3+(-3)λ^4+λ^5

    Real roots of f
    3
    -√17
    √17

    Complex roots of f
    3
    -i
    i
    -√17
    √17

    2-adic roots of f
    ...000011
    ...101001
    ...010111

    3-adic roots of f
    ...000010

    Roots of f in Q[i]
    (3)1
    i
    (-1)i
*/

p-Adic Algebraic Numbers

Isolated algebraic numbers in the \(p\)-adic fields \(\mathbb{Q}_p\).

Taking Square Roots

It is possible to take square roots of \(p\)-adic algebraic numbers by calling .square_roots.

use algebraeon::nzq::{Natural, Rational};
use algebraeon::rings::isolated_algebraic::PAdicAlgebraic;
use algebraeon::rings::{polynomial::*, structure::*};

let two_adic = PAdicAlgebraic::structure(Natural::from(2u32));
for x in two_adic
    .rational_extension()
    .all_roots(&Polynomial::<Rational>::from_str("(x - 3) * (x^2 - 17)", "x").unwrap())
{
    println!("{} is_square={}", x, two_adic.is_square(&x));
    if let Some((r1, r2)) = two_adic.square_roots(&x) {
        println!("    sqrt = {} and {}", r1, r2);
    }
}

/*
Output:
    ...000011 is_square=false
    ...101001 is_square=true
        sqrt = ...101101 and ...010011
    ...010111 is_square=false
*/

Ideals in Algebraic Rings of Integers

Factoring example

use algebraeon::rings::polynomial::PolynomialFromStr;
use algebraeon::{
    nzq::*,
    rings::{polynomial::Polynomial, structure::*},
};

// Construct the ring of integers Z[i]
let anf = Polynomial::<Rational>::from_str("x^2+1", "x")
    .unwrap()
    .algebraic_number_field()
    .unwrap();
let roi = anf.compute_ring_of_integers();

// The ideal (27i - 9) in Z[i]
let ideal = roi.ideals().principal_ideal(
    &roi.try_anf_to_roi(&Polynomial::from_str("27*x-9", "x").unwrap())
        .unwrap(),
);

// Factor the ideal
for (prime_ideal, power) in roi
    .ideals()
    .factorizations()
    .into_powers(roi.ideals().factor_ideal(&ideal).unwrap())
{
    println!("power = {power} prime_ideal_factor = {:?}", prime_ideal);
}

// There's not yet a nice way to print ideals so the output is messy
// But it prints the following factorization into primes
// ideal = (1+i) * (1+2i) * (3)^2