add simplfication/ canonicalization to FunctionFields

[fieker]

fixes Tobias' is_irreducible problem:

julia> Qt, t = rational_function_field(QQ, :t)
julia> Qtx, x = Qt[:x]
julia> f = x^4 + 4*(t^12-t)*x
julia> K = splitting_field(f)
julia> R, s = K[:s]
julia> r = roots(K, f)[1] #the root might vary
(1//72*b^4 + (-1//2*t^12 + 1//2*t)*b)//(t^12 - t)

julia> g = s^2 - r^3 - t^12 + t
s^2 + 3*t^12 - 3*t

julia> is_irreducible(g) #this kills time and memory without the fix

I do not really know how to test this...it does not change the mathematics, but the performance - in particular in functions not available in AA...

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 88.12%. Comparing base (3355a14) to head (7694dc7).

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #2393      +/-   ##
==========================================
- Coverage   88.13%   88.12%   -0.01%     
==========================================
  Files         128      128              
  Lines       32886    32890       +4     
==========================================
+ Hits        28983    28984       +1     
- Misses       3903     3906       +3     

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[Tobias271828]

@fieker Thank you very much for this helpful PR!

[lgoettgens]

Why only do this if the numerator is not zero? In the case of it being zero, couldn't we canonicalize even more by setting the denominator to 1?

[lgoettgens]

Suggested change

	c = inv(content(den))
	num *= c
	den *= c
	c = content(den)
	num = divexact(num, c)
	den = divexact(den, c)

A long time ago someone (you) told me that I should not use inv when doing division :)

[fieker]

On Fri, Apr 17, 2026 at 06:04:06AM -0700, Lars Göttgens wrote: @lgoettgens commented on this pull request. > @@ -1088,6 +1088,11 @@ mutable struct FunctionFieldElem{T <: FieldElement} <: AbstractAlgebra.FieldElem parent::FunctionField{T} function FunctionFieldElem{T}(R::FunctionField{T}, num::Poly{S}, den::S) where {T <: FieldElement, S <: PolyRingElem{T}} + if !iszero(num) #normalize the denominator Why only do this if the numerator is not zero? In the case of it being zero, couldn't we canonicalize even more by setting the denominator to 1?

Possibly. But hopefully in this case the denominator is not looked at at all

…

> + c = inv(content(den)) + num *= c + den *= c ```suggestion c = content(den) num = divexact(num, c) den = divexact(den, c) ``` A long time ago someone (you) told me that I should not use `inv` when doing division :) -- Reply to this email directly or view it on GitHub: #2393?email_source=notifications&email_token=AA36CVYWFAWCRHYHXQKISSD4WITUNA5CNFSNUABKM5UWIORPF5TWS5BNNB2WEL2QOVWGYUTFOF2WK43UKJSXM2LFO4XTIMJSHA4TQMRQGI22M4TFMFZW63VHNVSW45DJN5XKKZLWMVXHJPLQOJPXEZLWNFSXOX3ON52GSZTJMNQXI2LPNZZV6Y3MNFRWW#pullrequestreview-4128982025 You are receiving this because you were mentioned. Message ID: ***@***.***>