Polynomial ring realization.
(v1.8) pkg> add SimplePolynomialRingThis package supports Julia 1.8 and later.
julia> using SimplePolynomialRingPolynomial could be constructed from Julia Expr or String.
julia> a = Pℤ{3, :x}(:(1 + x + 2x^2))
1 + x + 2x^2
julia> b = Pℤ{3, :x}("1 + x + 2x^2")
1 + x + 2x^2As polynom's coefficients are modulo P, then the following polynomials are the same.
julia> c = Pℤ{3}(2)
2
julia> d = Pℤ{3}(5)
2
julia> c == d
truejulia> e = Pℤ₂([1,0,0,1,1,0,1]) # alias for `Pℤ{2, :x}`
1 + x^3 + x^4 + x^6Operaions with two polynomials.
julia> p = Pℤ{3, :x}(:(1 + 2x + x^3 + 2x^5))
1 + 2x + x^3 + 2x^5
julia> q = Pℤ{3, :x}(:(1 + x + 2x^2))
1 + x + 2x^2
julia> p + q
2 + 2x^2 + x^3 + 2x^5
julia> p - q
x + x^2 + x^3 + 2x^5
julia> p * q
1 + x^2 + 2x^3 + x^4 + x^5 + 2x^6 + x^7
julia> p ÷ q
(2 + x + x^2 + x^3, 2 + 2x)
julia> p / q
2 + x + x^2 + x^3
julia> p % q
2 + 2xOperations with a polynomial and a number.
julia> p = Pℤ{3, :x}(:(1 + 2x + x^3 + 2x^5))
1 + 2x + x^3 + 2x^5
julia> p * 5
2 + x + 2x^3 + x^5
julia> p ^ 5
1 + x + x^2 + x^3 + x^6 + 2x^7 + x^9 + 2x^10 + 2x^12 + x^14 + 2x^15 + 2x^16 + x^18 + 2x^20 + 2x^23 + 2x^25Extended GCD.
julia> p = Pℤ{3, :x}(:(1 + 2x + x^3 + 2x^5))
1 + 2x + x^3 + 2x^5
julia> q = Pℤ{3, :x}(:(1 + x + 2x^2))
1 + x + 2x^2
julia> gcdx(p, q)
(2, 2 + 2x, 2x^2 + 2x^3 + x^4)Factorization and primality test.
julia> p = Pℤ{3, :x}(:(1 + 2x + x^3 + 2x^5))
1 + 2x + x^3 + 2x^5
julia> isprime(p)
false
julia> factor(p)
Dict{Pℤ{3, :x}, Int64} with 2 entries:
2 + x => 2
1 + x + x^2 + 2x^3 => 1Polynomial generation: all possible and prime.
julia> generate(3, 3)
54-element Vector{Pℤ{3, :x}}:
x^3
1 + x^3
2 + x^3
x + x^3
1 + x + x^3
2 + x + x^3
2x + x^3
1 + 2x + x^3
2 + 2x + x^3
x^2 + x^3
1 + x^2 + x^3
⋮
2 + 2x + x^2 + 2x^3
2x^2 + 2x^3
1 + 2x^2 + 2x^3
2 + 2x^2 + 2x^3
x + 2x^2 + 2x^3
1 + x + 2x^2 + 2x^3
2 + x + 2x^2 + 2x^3
2x + 2x^2 + 2x^3
1 + 2x + 2x^2 + 2x^3
2 + 2x + 2x^2 + 2x^3
julia> primes(3, 3)
28-element Vector{Pℤ{3, :x}}:
x
1 + x
2 + x
2x
1 + 2x
2 + 2x
1 + x^2
2 + x + x^2
2 + 2x + x^2
2 + 2x^2
1 + x + 2x^2
⋮
1 + x + 2x^2 + x^3
2 + 2x + 2x^2 + x^3
1 + x + 2x^3
2 + x + 2x^3
2 + x^2 + 2x^3
1 + x + x^2 + 2x^3
2 + 2x + x^2 + 2x^3
1 + 2x^2 + 2x^3
2 + x + 2x^2 + 2x^3
1 + 2x + 2x^2 + 2x^3
julia> irreducible(3, 3)
16-element Vector{Pℤ{3, :x}}:
1 + 2x + x^3
2 + 2x + x^3
2 + x^2 + x^3
2 + x + x^2 + x^3
1 + 2x + x^2 + x^3
1 + 2x^2 + x^3
1 + x + 2x^2 + x^3
2 + 2x + 2x^2 + x^3
1 + x + 2x^3
2 + x + 2x^3
2 + x^2 + 2x^3
1 + x + x^2 + 2x^3
2 + 2x + x^2 + 2x^3
1 + 2x^2 + 2x^3
2 + x + 2x^2 + 2x^3
1 + 2x + 2x^2 + 2x^3