Hi guys,
I know that for a variable $x$ in $GF(2)$, $x^2 = x$, and $2x=0$.
How do I simplify a polynomial expression in $GF(2)$ in the Sage interface?
For example, I should obtain
$$(a+b+1)^2=a^2+b^2+1+2a+2b+2ab=a+b+1$$
The simplest is to work over boolean polynomials
sage: B.<a,b> = BooleanPolynomialRing(2)
sage: (a+b+1)^2
a + b + 1
But you can also use the more general quotient of polynomial rings (notice that in GF(2) a equals -a)
sage: R.<a,b> = PolynomialRing(GF(2), 'a,b')
sage: Rbar = R.quotient([a^2+a, b^2+b])
sage: abar,bbar = Rbar.gens()
sage: abar
abar
sage: (abar+bbar+1)^2
abar + bbar + 1
I'm not sure why you think that x^2=x and Sage gives
sage: R.<a,b> = PolynomialRing(GF(2), 'a,b')
sage: (a+b+1)^2
a^2 + b^2 + 1
The question is how to manipulate expressions involving variables which are assumed to live in GF(2). In this context, x, a, b, etc. are thought of as elements in GF(2), not as indeterminates of polynomial rings over in GF(2). The question mentions polynomial expressions in such variables, but this does not mean formal polynomials.
Asked: 2015-02-16 03:45:32 +0100
Seen: 449 times
Last updated: Feb 16 '15
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