noether 0.3.0

//! Field theory structures.
//!
//! This module defines traits for algebraic structures related to fields,
//! including Euclidean domains, fields, and field extensions.

use crate::groups::MultiplicativeAbelianGroup;
use crate::rings::PrincipalIdealDomain;
use crate::spaces::VectorSpace;
use num_traits::{Euclid, One, Zero};

/// Represents a Euclidean Domain, an integral domain with a Euclidean function.
///
/// # Mathematical Definition
/// A Euclidean Domain (R, +, ·, φ) is a principal ideal domain equipped with a
/// Euclidean function φ: R\{0} → ℕ₀ that satisfies certain properties.
///
/// # Formal Definition
/// Let (R, +, ·) be an integral domain and φ: R\{0} → ℕ₀ a function. R is a Euclidean domain if:
/// 1. ∀a, b ∈ R, b ≠ 0, ∃!q, r ∈ R : a = bq + r ∧ (r = 0 ∨ φ(r) < φ(b)) (Division with Remainder)
/// 2. ∀a, b ∈ R\{0} : φ(a) ≤ φ(ab) (Multiplicative Property)
pub trait EuclideanDomain: PrincipalIdealDomain + Euclid {}

/// Represents a Field, a commutative ring where every non-zero element has a multiplicative inverse.
///
/// # Mathematical Definition
/// A field (F, +, ·) is a commutative ring where:
/// - Every non-zero element has a multiplicative inverse
///
/// # Formal Definition
/// Let (F, +, ·) be a field. Then:
/// 1. (F, +, ·) is a commutative ring
/// 2. ∀ a ∈ F, a ≠ 0, ∃ a⁻¹ ∈ F, a · a⁻¹ = a⁻¹ · a = 1 (multiplicative inverse)
pub trait Field: EuclideanDomain + MultiplicativeAbelianGroup {}

/// Represents a Finite Field, a field with a finite number of elements.
///
/// # Mathematical Definition
/// A finite field is a field with a finite number of elements.
///
/// # Properties
/// - The number of elements is always a prime power p^n
pub trait FiniteField: Field {
    ///Allows for an arbitrary size representation without specificing type
    type ScalarType: Clone + PartialOrd + Zero + One;

    /// Returns the characteristic of the field.
    fn characteristic() -> u64;

    /// Returns the number of elements in the field.
    fn order() -> u64;
}

/// Represents an Ordered Field, a field with a total order compatible with its operations.
///
/// # Mathematical Definition
/// An ordered field is a field equipped with a total order ≤ where:
/// - If a ≤ b then a + c ≤ b + c for all c
/// - If 0 ≤ a and 0 ≤ b then 0 ≤ a · b
pub trait OrderedField: Field + PartialOrd {}

/// Represents a Real Field, a complete ordered field.
///
/// # Mathematical Definition
/// A real field is an ordered field that is Dedekind-complete:
/// - Every non-empty subset with an upper bound has a least upper bound
pub trait RealField: OrderedField {}

/// Represents a Field Extension.
///
/// # Mathematical Definition
/// A field extension L/K is a field L that contains K as a subfield.
///
/// # Properties
/// - L is a field
/// - K is a subfield of L
/// - L is a vector space over K
pub trait FieldExtension: Field + VectorSpace<Self::BaseField> {
    /// The base field of this extension.
    type BaseField: Field;

    /// Returns the degree of the field extension.
    fn degree() -> usize;

    /// Computes the trace of an element.
    fn trace(&self) -> Self::BaseField;

    /// Computes the norm of an element.
    fn norm(&self) -> Self::BaseField;
}

/// Represents a Tower of Field Extensions.
///
/// # Mathematical Definition
/// A tower of field extensions is a sequence of fields K = F₀ ⊂ F₁ ⊂ ... ⊂ Fₙ = L
/// where each Fᵢ₊₁/Fᵢ is a field extension.
///
/// # Properties
/// - Each level is a field extension of the previous level
/// - The composition of the extensions forms the overall extension L/K
/// - The degree of L/K is the product of the degrees of each extension in the tower
pub trait FieldExtensionTower: FieldExtension {
    /// Returns the number of extensions in the tower.
    fn tower_height() -> usize;

    /// Returns the degree of the i-th extension in the tower.
    fn extension_degree(i: usize) -> usize;
}

// Blanket implementations
impl<T: PrincipalIdealDomain + Euclid> EuclideanDomain for T {}
impl<T: EuclideanDomain + MultiplicativeAbelianGroup> Field for T {}
impl<T: Field + PartialOrd> OrderedField for T {}

// RealField
// Note: This cannot be implemented as a blanket impl because it requires knowledge about completeness

// FieldExtension
// Note: This cannot be implemented as a blanket impl because it requires specific knowledge about the base field and extension structure

// FieldExtensionTower
// Note: This cannot be implemented as a blanket impl because it requires specific knowledge about the tower structure

#[cfg(test)]
mod tests {
    use super::*;
    use crate::operations::{
        AssociativeAddition, AssociativeMultiplication, CommutativeAddition,
        CommutativeMultiplication, Distributive,
    };
    use num_traits::{Inv, One, Zero};
    use std::cmp::Ordering;
    use std::ops::{
        Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Rem, RemAssign, Sub, SubAssign,
    };

    // Implement a simple finite field (GF(5) - integers modulo 5)
    #[derive(Debug, Clone, Copy, PartialEq, Eq)]
    struct GF5(u8);

    impl GF5 {
        // Create a new element, ensuring it is normalized into the range [0, 4]
        fn new(value: u8) -> Self {
            Self(value % 5)
        }
    }

    // Additive operations for GF5
    impl Add for GF5 {
        type Output = Self;

        fn add(self, rhs: Self) -> Self::Output {
            Self::new(self.0 + rhs.0)
        }
    }

    impl AddAssign for GF5 {
        fn add_assign(&mut self, rhs: Self) {
            *self = *self + rhs;
        }
    }

    impl Sub for GF5 {
        type Output = Self;

        fn sub(self, rhs: Self) -> Self::Output {
            // Add 5 to handle the case where self.value < rhs.value
            Self::new(self.0 + 5 - rhs.0)
        }
    }

    impl SubAssign for GF5 {
        fn sub_assign(&mut self, rhs: Self) {
            *self = *self - rhs;
        }
    }

    impl Neg for GF5 {
        type Output = Self;

        fn neg(self) -> Self::Output {
            if self.0 == 0 {
                Self(0) // -0 = 0
            } else {
                Self(5 - self.0)
            }
        }
    }

    impl Zero for GF5 {
        fn zero() -> Self {
            Self(0)
        }

        fn is_zero(&self) -> bool {
            self.0 == 0
        }
    }

    // Multiplicative operations for GF5
    impl Mul for GF5 {
        type Output = Self;

        fn mul(self, rhs: Self) -> Self::Output {
            Self::new(self.0 * rhs.0)
        }
    }

    impl MulAssign for GF5 {
        fn mul_assign(&mut self, rhs: Self) {
            *self = *self * rhs;
        }
    }

    impl One for GF5 {
        fn one() -> Self {
            Self(1)
        }
    }

    // Multiplicative inverses in GF(5)
    impl Inv for GF5 {
        type Output = Self;

        fn inv(self) -> Self::Output {
            match self.0 {
                0 => panic!("Division by zero in GF(5)"),
                1 => Self(1), // 1*1 ≡ 1 (mod 5)
                2 => Self(3), // 2*3 ≡ 6 ≡ 1 (mod 5)
                3 => Self(2), // 3*2 ≡ 6 ≡ 1 (mod 5)
                4 => Self(4), // 4*4 ≡ 16 ≡ 1 (mod 5)
                _ => unreachable!("All values should be normalized to 0-4"),
            }
        }
    }

    impl Div for GF5 {
        type Output = Self;

        fn div(self, rhs: Self) -> Self::Output {
            if rhs.0 == 0 {
                panic!("Division by zero in GF(5)");
            }
            self * rhs.inv()
        }
    }

    impl DivAssign for GF5 {
        fn div_assign(&mut self, rhs: Self) {
            *self = *self / rhs;
        }
    }

    impl Rem for GF5 {
        type Output = Self;

        fn rem(self, rhs: Self) -> Self::Output {
            if rhs.0 == 0 {
                panic!("Remainder by zero in GF(5)");
            }
            Self(self.0 % rhs.0)
        }
    }

    impl RemAssign for GF5 {
        fn rem_assign(&mut self, rhs: Self) {
            if rhs.0 == 0 {
                panic!("Remainder by zero in GF(5)");
            }
            *self = Self(self.0 % rhs.0);
        }
    }

    // Implement Euclid trait for GF5
    impl Euclid for GF5 {
        fn div_euclid(&self, v: &Self) -> Self {
            if v.0 == 0 {
                panic!("Division by zero in GF(5)");
            }
            *self / *v
        }

        fn rem_euclid(&self, v: &Self) -> Self {
            if v.0 == 0 {
                panic!("Remainder by zero in GF(5)");
            }
            Self(self.0 % v.0)
        }
    }

    // Marker traits for GF5
    impl CommutativeAddition for GF5 {}
    impl AssociativeAddition for GF5 {}
    impl CommutativeMultiplication for GF5 {}
    impl AssociativeMultiplication for GF5 {}
    impl Distributive for GF5 {}

    impl PartialOrd for GF5 {
        fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
            self.0.partial_cmp(&other.0)
        }
    }

    // Implement FiniteField for GF5
    impl FiniteField for GF5 {
        type ScalarType = u8;

        fn characteristic() -> u64 {
            5
        }

        fn order() -> u64 {
            5
        }
    }

    // Implement a simple ordered field based on rational numbers
    #[derive(Debug, Clone, Copy, PartialEq)]
    struct Rational {
        num: i32,
        den: i32,
    }

    impl Rational {
        // Create a new rational number in simplified form
        fn new(num: i32, den: i32) -> Self {
            if den == 0 {
                panic!("Denominator cannot be zero");
            }

            // Ensure the denominator is positive
            let (num, den) = if den < 0 { (-num, -den) } else { (num, den) };

            // Simplify the fraction using GCD
            let gcd = Self::gcd(num.abs(), den);
            Self {
                num: num / gcd,
                den: den / gcd,
            }
        }

        // Compute the greatest common divisor using Euclidean algorithm
        fn gcd(a: i32, b: i32) -> i32 {
            if b == 0 {
                a
            } else {
                Self::gcd(b, a % b)
            }
        }
    }

    // Additive operations for Rational
    impl Add for Rational {
        type Output = Self;

        fn add(self, rhs: Self) -> Self::Output {
            Self::new(self.num * rhs.den + rhs.num * self.den, self.den * rhs.den)
        }
    }

    impl AddAssign for Rational {
        fn add_assign(&mut self, rhs: Self) {
            *self = *self + rhs;
        }
    }

    impl Sub for Rational {
        type Output = Self;

        fn sub(self, rhs: Self) -> Self::Output {
            Self::new(self.num * rhs.den - rhs.num * self.den, self.den * rhs.den)
        }
    }

    impl SubAssign for Rational {
        fn sub_assign(&mut self, rhs: Self) {
            *self = *self - rhs;
        }
    }

    impl Neg for Rational {
        type Output = Self;

        fn neg(self) -> Self::Output {
            Self::new(-self.num, self.den)
        }
    }

    impl Zero for Rational {
        fn zero() -> Self {
            Self::new(0, 1)
        }

        fn is_zero(&self) -> bool {
            self.num == 0
        }
    }

    // Multiplicative operations for Rational
    impl Mul for Rational {
        type Output = Self;

        fn mul(self, rhs: Self) -> Self::Output {
            Self::new(self.num * rhs.num, self.den * rhs.den)
        }
    }

    impl MulAssign for Rational {
        fn mul_assign(&mut self, rhs: Self) {
            *self = *self * rhs;
        }
    }

    impl One for Rational {
        fn one() -> Self {
            Self::new(1, 1)
        }
    }

    impl Inv for Rational {
        type Output = Self;

        fn inv(self) -> Self::Output {
            if self.num == 0 {
                panic!("Division by zero");
            }
            Self::new(self.den, self.num)
        }
    }

    impl Div for Rational {
        type Output = Self;

        fn div(self, rhs: Self) -> Self::Output {
            if rhs.num == 0 {
                panic!("Division by zero");
            }
            Self::new(self.num * rhs.den, self.den * rhs.num)
        }
    }

    impl DivAssign for Rational {
        fn div_assign(&mut self, rhs: Self) {
            *self = *self / rhs;
        }
    }

    impl Rem for Rational {
        type Output = Self;

        fn rem(self, rhs: Self) -> Self::Output {
            if rhs.num == 0 {
                panic!("Remainder by zero");
            }
            // a % b = a - (a / b).floor() * b
            let quotient = (self.num as f64 * rhs.den as f64) / (self.den as f64 * rhs.num as f64);
            let whole_part = Self::new(quotient.floor() as i32, 1);
            self - whole_part * rhs
        }
    }

    impl RemAssign for Rational {
        fn rem_assign(&mut self, rhs: Self) {
            *self = *self % rhs;
        }
    }

    // Euclidean functions for Rational
    impl Euclid for Rational {
        fn div_euclid(&self, v: &Self) -> Self {
            if v.num == 0 {
                panic!("Division by zero");
            }
            // For rationals, we define Euclidean division as rounding down
            let quotient = (self.num as f64 * v.den as f64) / (self.den as f64 * v.num as f64);
            Self::new(quotient.floor() as i32, 1)
        }

        fn rem_euclid(&self, v: &Self) -> Self {
            if v.num == 0 {
                panic!("Remainder by zero");
            }
            let quotient = self.div_euclid(v);
            *self - quotient * *v
        }
    }

    // Partial ordering for Rational
    impl PartialOrd for Rational {
        fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
            // Compare a/b and c/d by computing a*d vs b*c
            let left = self.num as i64 * other.den as i64;
            let right = self.den as i64 * other.num as i64;
            left.partial_cmp(&right)
        }
    }

    // Marker traits for Rational
    impl CommutativeAddition for Rational {}
    impl AssociativeAddition for Rational {}
    impl CommutativeMultiplication for Rational {}
    impl AssociativeMultiplication for Rational {}
    impl Distributive for Rational {}

    #[test]
    fn test_euclidean_domain() {
        fn assert_is_euclidean_domain<T: EuclideanDomain>(_: &T) {}

        // Test GF5 implements EuclideanDomain
        assert_is_euclidean_domain(&GF5::new(2));

        // Test Rational implements EuclideanDomain
        assert_is_euclidean_domain(&Rational::new(3, 4));

        // Test primitive float implements EuclideanDomain
        assert_is_euclidean_domain(&std::f64::consts::PI);

        // Test euclidean division in GF5
        let a = GF5::new(7); // 7 mod 5 = 2
        let b = GF5::new(3); // 3

        // In GF5: 2 / 3 = 2 * inverse(3) = 2 * 2 = 4
        assert_eq!(a.div_euclid(&b), GF5::new(4));

        // In GF5: 2 % 3 = 2 (since 3 * 4 + 2 = 14 ≡ 4 mod 5)
        // But this is more complex - for simplicity just ensuring it's defined
        let _ = a.rem_euclid(&b);

        // Test euclidean division in Rational
        let p = Rational::new(7, 2); // 7/2 = 3.5
        let q = Rational::new(2, 1); // 2

        // 3.5 divided by 2 gives quotient 1 (floor of 1.75) and remainder 1.5
        assert_eq!(p.div_euclid(&q), Rational::new(1, 1));
        assert_eq!(p.rem_euclid(&q), Rational::new(3, 2));
    }

    #[test]
    fn test_field() {
        fn assert_is_field<T: Field>(_: &T) {}

        // Test GF5 implements Field
        assert_is_field(&GF5::new(2));

        // Test Rational implements Field
        assert_is_field(&Rational::new(3, 4));

        // Test primitive float implements Field
        assert_is_field(&std::f64::consts::PI);

        // Test field properties in GF5
        let a = GF5::new(2);
        let b = GF5::new(3);

        // Addition is commutative
        assert_eq!(a + b, b + a);

        // Multiplication is commutative
        assert_eq!(a * b, b * a);

        // Distributivity
        let c = GF5::new(4);
        assert_eq!(a * (b + c), a * b + a * c);

        // Multiplicative inverse
        assert_eq!(a * a.inv(), GF5::one());
        assert_eq!(b * b.inv(), GF5::one());

        // Division
        assert_eq!(a / b, a * b.inv());

        // Test field properties in Rational
        let p = Rational::new(3, 4); // 3/4
        let q = Rational::new(2, 5); // 2/5

        // Addition is commutative
        assert_eq!(p + q, q + p);

        // Multiplication is commutative
        assert_eq!(p * q, q * p);

        // Distributivity
        let r = Rational::new(1, 2); // 1/2
        assert_eq!(p * (q + r), p * q + p * r);

        // Multiplicative inverse
        assert_eq!(p * p.inv(), Rational::one());
        assert_eq!(q * q.inv(), Rational::one());

        // Division
        assert_eq!(p / q, p * q.inv());
    }

    #[test]
    fn test_finite_field() {
        fn assert_is_finite_field<T: FiniteField>(_: &T) {}

        // Test GF5 implements FiniteField
        assert_is_finite_field(&GF5::new(2));

        // Test field properties
        assert_eq!(GF5::characteristic(), 5);
        assert_eq!(GF5::order(), 5);

        // Test all field elements
        let elements: Vec<GF5> = (0..5).map(GF5::new).collect();

        // Verify closure under operations
        for &a in &elements {
            for &b in &elements {
                if !b.is_zero() {
                    let sum = a + b;
                    let product = a * b;
                    let quotient = a / b;

                    // Results should be in the field
                    assert!(elements.contains(&sum));
                    assert!(elements.contains(&product));
                    assert!(elements.contains(&quotient));
                }
            }
        }

        // Verify that every non-zero element has a unique multiplicative inverse
        let mut seen_inverses = Vec::new();
        for &a in &elements {
            if !a.is_zero() {
                let a_inv = a.inv();
                assert!(!seen_inverses.contains(&a_inv));
                seen_inverses.push(a_inv);

                // Verify inverse property
                assert_eq!(a * a_inv, GF5::one());
            }
        }
    }

    #[test]
    fn test_ordered_field() {
        fn assert_is_ordered_field<T: OrderedField>(_: &T) {}

        // Test Rational implements OrderedField
        assert_is_ordered_field(&Rational::new(3, 4));

        // Test primitive float implements OrderedField
        assert_is_ordered_field(&std::f64::consts::PI);

        // Test ordering properties of Rational
        let a = Rational::new(3, 4); // 3/4
        let b = Rational::new(2, 3); // 2/3
        let c = Rational::new(1, 2); // 1/2

        // Basic comparison
        assert!(a > b && b > c);
        assert!(a > c);

        // Order compatibility with addition
        let d = Rational::new(1, 10); // 1/10
        assert!(a > b);
        assert!(a + d > b + d);

        // Order compatibility with positive multiplication
        let pos = Rational::new(2, 1); // 2
        assert!(a > c);
        assert!(a * pos > c * pos);

        // Negative numbers and order
        let neg = Rational::new(-2, 1); // -2
        assert!(a * neg < c * neg); // -3/2 < -1
    }

    #[test]
    fn test_field_axioms() {
        // Test the field axioms using GF5
        let elements: Vec<GF5> = (0..5).map(GF5::new).collect();
        let non_zero: Vec<GF5> = elements.iter().filter(|e| !e.is_zero()).cloned().collect();

        let zero = GF5::zero();
        let one = GF5::one();

        // 1. Additive identity: a + 0 = a
        for &a in &elements {
            assert_eq!(a + zero, a);
        }

        // 2. Additive inverse: a + (-a) = 0
        for &a in &elements {
            assert_eq!(a + (-a), zero);
        }

        // 3. Multiplicative identity: a * 1 = a
        for &a in &elements {
            assert_eq!(a * one, a);
        }

        // 4. Multiplicative inverse: a * a^(-1) = 1 for all a ≠ 0
        for &a in &non_zero {
            assert_eq!(a * a.inv(), one);
        }

        // 5. Addition is associative: (a + b) + c = a + (b + c)
        for &a in &elements {
            for &b in &elements {
                for &c in &elements {
                    assert_eq!((a + b) + c, a + (b + c));
                }
            }
        }

        // 6. Addition is commutative: a + b = b + a
        for &a in &elements {
            for &b in &elements {
                assert_eq!(a + b, b + a);
            }
        }

        // 7. Multiplication is associative: (a * b) * c = a * (b * c)
        for &a in &elements {
            for &b in &elements {
                for &c in &elements {
                    assert_eq!((a * b) * c, a * (b * c));
                }
            }
        }

        // 8. Multiplication is commutative: a * b = b * a
        for &a in &elements {
            for &b in &elements {
                assert_eq!(a * b, b * a);
            }
        }

        // 9. Distributivity: a * (b + c) = a * b + a * c
        for &a in &elements {
            for &b in &elements {
                for &c in &elements {
                    assert_eq!(a * (b + c), a * b + a * c);
                }
            }
        }
    }
}