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Nemocas / AbstractAlgebra.jl

Generic abstract algebra functionality in pure Julia (no C dependencies)
https://nemocas.github.io/AbstractAlgebra.jl/dev/index.html
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Documentation of "Canonicalisation" #1153

Open ulthiel opened 2 years ago

ulthiel commented 2 years ago

I had to read the paragraph of Canonicalisation several times to understand what it does. Can we improve this?

Here's a suggestion. Consider the associates relation ∼ on R, i.e. x ∼ y iff there is a unit u such that y = xu^{-1}. We fix a system ℛ of representatives. Given an element x ∈ R, its canonical unit is the unit u such that xu^{-1} ∈ ℛ.

For a field we would fix ℛ = {0,1} (and then the canonical unit of any non-zero u is u), for ℤ we take ℛ = ℕ, for K[X] we take monic polynomials and 0, for Gaussian integers the first quadrant, etc.

ulthiel commented 2 years ago

Oh, I mentioned this long ago in #552 but forgot about it again...