I'm new to sage and was introduced to it this semester in my
Cryptography course as being a python friendly alternative to Pari/GP.
So playing around with it I was factoring polynomials over GF(2) and
trying to make a quick function to generate random polynomials and
return if they were reducible, without me actually visually inspecting
the factorization. For the sake of giving an example I'll just define
a couple of polynomials:
sage: R.<x> = PolynomialRing(GF(2))
sage: p = x^30 + x^21 + 1
sage: q = x^30 + x^14 + 1
sage: fac_p = factor(p); fac_p
x^30 + x^21 + 1
sage: fac_q = factor(q); fac_q
(x^15 + x^7 + 1)^2
Now here is the interesting bit:
sage: fac_q == q # should return False
False
sage: fac_p == p # would expect True
False
What!? But p is irreducible, its factorization should be itself. After
much digging around and thinking about things from a programmers
perspective rather than a mathematics perspective the idea that
factorizations are lists of factors and why comparing a factorization
to a polynomial doesn't work. However, I would contend that the
comparison does in fact make intuitive sense. So, perhaps comparison
testing for Factorizations and Polynomials should be a bit more
"intelligent" or at least provide some easy way of comparing them in
the intuitive sense of compare. After all this does work:
sage: fac_p[0][0] == p
True
And that would only hold true (as far as I know) if the polynomial was
irreducible.
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