I have to solve the following problem with GAP:
Given R = $\mathbb{Z}_2[x]/(x^5+x^4+1)$, I have to compute the kernel of $f:R \to R$ given by $f(p)=p^2-p$.
To do that I would like to know if f is an homomorphism.
- Is there some function in GAP to solve this?
- How can I compute if this is an homomorfisms?
What I did: If it were a field it would be great because the Frobenius Endomorfism is well understood, but this is not a field since $x^5+x^4+1$ is not irreducible over $\mathbb{Z}_2[x]$.
On the other hand, I programmed the following:
x:=Indeterminate(ZmodnZ(2),"x");
modulo:=x^5+x^4+1;
P:=PolynomialRing(ZmodnZ(2),"x");
R:=P/Ideal(P,[modulo]);
nucleo:=Filtered(Elements(R), x->x^2=x);
which give that the kernel is:
[ 0*(1), (x2)+(x3)+(x4), (1), (1)+(x2)+(x3)+(x4) ]
- Is the code above correct to solve the problem?
What you did is correct for getting the result. It computes (in a brute-force way) the kernel, assuming the map is a homomorphism. You migth want to observe that it is a 2-dimensional vector space. To show that it is a homomorphism it probably is easiest to use a theoretical argument and show that the map is a homomorphism on $Z_2$ and extends to the full polynomial ring. (This will be easier, as the polynomial ring is a free cyclic algebra.)