Show that $f:\Bbb{F}_3[x]\to\Bbb{F}_3[x]$ given by $a(x)\mapsto a(x)^3$ is a group homomorphism.

Question

Define $f:\Bbb{F}_3[x]\to\Bbb{F}_3[x]$ by $$a(x)\mapsto a(x)^3$$ How do I show that $f$ is a group homomorphism?

Answer 1

Let $a,b\in\mathbb{F}_3[x]$ than $$ f(a+b) = (a + b)^3 = a^3 + 3a^2 b + 3 ab^2+b^3 = a^3+b^3 $$ because $3=0\in\mathbb{F}_3$, so $f(a+b) = f(a) + f(b)$. This means $f$ is a group homomorphism.