Apparent bug when factoring over a finite field

asked 2025-01-27 08:18:34 +0200

tabesbridges
1

updated 2025-02-06 10:20:29 +0200

FrédéricC

5466 ●4 ●45 ●122

The code below constructs the finite field F of order 7, defines an irreducible polynomial p over F, and then constructs the associated field extension K (the "splitting field of p") before checking that the polynomial in fact splits over K. When I run this code (for instance in a Jupyter notebook or on SageMathCell) I get an extensive error message pertaining to the factor() method. If I modify the code slightly, changing the exponents in both p and f from 2 to 3, the code executes without issue. Could this be a bug, or have I forgotten something about basic field theory that is causing this?

F = Integers(7)
P.<x> = F[]
p = x^2 + 2
K.<c> = F.extension(p)
S.<z> = K[]
f = z^2 + 2
factor(f)

answered 2025-01-27 22:01:51 +0200

Max Alekseyev

7126 ●10 ●60 ●175

updated 2025-01-27 22:03:50 +0200

Please update your version of Sage and note that ring Integers(7) as Sage's object technically is not the same as the finite field GF(7).

Check this out: https://sagecell.sagemath.org/?q=wmktnq

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Thanks for the answer; 10.4 is currently the highest version of Sage available on CoCalc, to which I am constrained for institutional reasons. I understand that Integers(7) is not technically the same object, but it has all of the necessary functionality (namely the field operations), and it works for higher degree extensions, so this answer is somewhat unsatisfying (or rather, is there a deeper fundamental reason within Sage that I should be avoiding this isomorphic object?).

tabesbridges ( 2025-01-27 23:20:05 +0200 )edit

Sagecell is running version 10.2, and so your 10.4 is good enough. Please note that your code with Integers(7) produces an error at Sagecell as it is not automatically understood as a field. Please just use GF(7) if you want to rely on the field functionality without issues.

Max Alekseyev ( 2025-01-27 23:46:19 +0200 )edit

Alternatively, you can define F = Integers(7).field() to make things work.

Max Alekseyev ( 2025-01-27 23:54:19 +0200 )edit

Clarification on Sagecell.

Emmanuel Charpentier ( 2025-01-31 08:05:13 +0200 )edit

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