# Math Help - idempotents

1. ## idempotents

What are three idempotents of the quotient ring $\mathbb{Q}[x]/(x^4+x^2)$?

So an element is of the form $a+x^2+x^4$ where $a \in \mathbb{Q}[x]$. So $(a+x^2+x^4)^2 = a+x^2+x^4$.

So one idempotent would be: $0$. The others would be just guess and check?

2. Originally Posted by Sampras
What are three idempotents of the quotient ring $\mathbb{Q}[x]/(x^4+x^2)$?

So an element is of the form $a+x^2+x^4$ where $a \in \mathbb{Q}[x]$. So $(a+x^2+x^4)^2 = a+x^2+x^4$.

So one idempotent would be: $0$. The others would be just guess and check?

Besides 0 I'd check 1 (!!) and $x^2+1$

Tonio

3. Originally Posted by Sampras
What are three idempotents of the quotient ring $\mathbb{Q}[x]/(x^4+x^2)$?

So an element is of the form $a+x^2+x^4$ where $a \in \mathbb{Q}[x]$. So $(a+x^2+x^4)^2 = a+x^2+x^4$.

So one idempotent would be: $0$. The others would be just guess and check?
use Chinese Remainder Theorem for rings:

since $x^2$ and $x^2+1$ are coprime, we have $\frac{\mathbb{Q}[x]}{} \cong \frac{\mathbb{Q}[x]}{} \times \frac{\mathbb{Q}[x]}{}.$ obviously for any two rings $R,S,$ an element $(a,b) \in R \times S$ is an idempotent iff $a$ and $b$ are idempotents.