I'm trying to understand how I show an ideal is a proper ideal. I think I am over complicating it, and would appreciate someone to help me understand.
As an example:
$I = (x^{4}+x^{2}-2x+1, x^{2}+x+10) ⊂ \Bbb{Q}[x]$
I know this is a principal ideal, I've used euclidean division to find this. How do I now show whether this is a proper ideal? Please try to be as clear as possible, I've been getting myself confused for hours now.
Thank you in advance.
As you pointed out, use the euclidean algorithm to find a $P\in\mathbb{Q}[x]$ so that $ I=(P) $.
If $P$ is not a constant, then $P+1$ is not in $I$. This is because if $Q=P+1$ where in $I$ then diving $P$ by $Q$ we obtain, $P=Q\cdot M+R$ where deg($R$)$<$deg(P), since $R=P-Q\cdot M$, $R\in I$. But since it's degree is strictly smaller than it's generator, $R=0$. Which is absurd.
Therefore $I$ is proper(because we found an element in $I$ not in $\mathbb{Q}[x]$)
Conversely, if $I$ is a constant, by Bezout theorem $\exists \lambda, \mu$ so that $\lambda\cdot(x^{4}+x^{2}-2x+1)$ $+$ $\mu\cdot(x^{2}+x+10)=1$, so $1\in\mathbb{Q}$ and so $I=\mathbb{Q}$