sage: k.<a> = GF(5^3)
....: Frob = k.frobenius_endomorphism()
....: S = OrePolynomialRing(k, Frob, 'x')
....: S
Ore Polynomial Ring in x over Finite Field in a of size 5^3 twisted by a |--> a^5
sage: x = S.gen()
sage: f = x*a + a*x^2+a*x^4
sage: g = (a^7+a)*x^2*(a+1)+ (a^3+1)*x + 1
sage: f
a*x^4 + a*x^2 + (2*a^2 + 4*a + 4)*x
sage: g
(3*a^2 + 2*a + 4)*x^2 + (2*a + 3)*x + 1
sage: q = f // g
sage: r = f % g
sage: f == g*q+r
False
sage: f == q*g+r
True
According to the documentation (1)
The operators // and % give respectively the quotient and the remainder of the right euclidean division
But the definition of the right euclidean division is (2)
Apparently the two docs are inconsistent. What is the correct definition of the right euclidean division? @xcaruso
According to Ore himself, the right-hand division of $F$ by $G$ is $F=QG+R$. So (2) is wrong.
According to the documentation (1)
But the definition of the right euclidean division is (2)
Apparently the two docs are inconsistent. What is the correct definition of the right euclidean division? @xcaruso
According to Ore himself, the right-hand division of$F$ by $G$ is $F=QG+R$ . So (2) is wrong.