# Thread: X^2 injective in a ring

1. ## X^2 injective in a ring

Let $A$ be a finite comutative ring . If $f:A->A$ , $f(x)=X^2$ is injective prove that :
a) $1+1=0$
b) $P(x)=x^{2n}+a$ is reductible for all $n \in N$ and for all $a \in A$

2. Convince yourself that $f(-1)=f(1)$ and it follows (because f is injective) that $-1=1$ or equivalently 1+1=0.

3. Part b) is a bit trickier.

f is injective so there exists $b \in A$ such that $f(b)=a$. Hence $a^{1/2}$ is exists.

Part a) shows that A has characteristic 2 and hence
$P(x)=x^{2n}+a=(x^n+a^{1/2})^2$

Ta da. P(x) is reducible