Is Julia lazy?

New to Julia
73

Don’t you need det, inv, etc. of a matrix for math educational purpose?

using GaloisFields, LinearAlgebra
GF7 = @GaloisField 7
B = GF7[1 2; 3 4]
det(B)

MethodError: no method matching abs(::𝔽₇)
...

inv(B)

MethodError: no method matching abs(::𝔽₇)
...

In the sense of my minimal definition example, minimality requires that inv and lu in LinearAlgebra.jl be available. I have tried to make the code small so that you can easily read it for using Julia in math education.

P.S. Iterator example (continuation of the minimal definition example above)

"""See https://docs.julialang.org/en/v1/manual/interfaces/"""
Base.iterate(Fp::Type{F{p, T}}) where {p, T<:Integer} = (zero(Fp), zero(T))
function Base.iterate(Fp::Type{F{p, T}}, state) where {p, T<:Integer}
    nextstate = state + 1
    nextstate < p ? (Fp(nextstate), nextstate) : nothing
end
Base.IteratorSize(Fp::Type{F{p, T}}) where {p, T<:Integer} = Base.HasLength()
Base.length(Fp::Type{F{p, T}}) where {p, T<:Integer} = p
Base.eltype(Fp::Type{F{p, T}}) where {p, T<:Integer} = Fp

squares(Fp) = Fp[x^2 for x in Fp]
squareroots(k, Fp) = Fp[x for x in Fp if x^2 == k]
@show(collect(F7), squares(F7), squareroots.(0:6, Ref(F7)));

Output:

collect(F7) = F{7, BigInt}[F7(0), F7(1), F7(2), F7(3), F7(4), F7(5), F7(6)]
squares(F7) = F{7, BigInt}[F7(0), F7(1), F7(4), F7(2), F7(2), F7(4), F7(1)]
squareroots.(0:6, Ref(F7)) = Vector{F{7, BigInt}}[[F7(0)], [F7(1), F7(6)], [F7(3), F7(4)], [], [F7(2), F7(5)], [], []]