1. ## Fews rings question

I have few questions about rings concept that i not really sure:

(a)Let $\displaystyle F$ be field. Every ideal in $\displaystyle F[x]$ is a prime ideal.

(b)Let $\displaystyle F$ be field. Every ideal in $\displaystyle F[x]$ is a principal ideal.(I got this one)

(c)If $\displaystyle \delta$ is Euclidean norm on Euclidean domain $\displaystyle D$ then $\displaystyle \delta(a)=\delta(b)$ if $\displaystyle a,b \in D$ are associates.

Please let me know is that correct, little bit of explain is ok. I'm not looking for a proof. Thank you

2. Originally Posted by kleenex
(a)Let $\displaystyle F$ be field. Every ideal in $\displaystyle F[x]$ is a prime ideal.
this is very false. for example, $\displaystyle <x^2>$ is not prime.

(b)Let $\displaystyle F$ be field. Every ideal in $\displaystyle F[x]$ is a principal ideal. (I got this one)
good! just note that in general every Euclidean domain is PID.

(c)If $\displaystyle \delta$ is Euclidean norm on Euclidean domain $\displaystyle D$ then $\displaystyle \delta(a)=\delta(b)$ if $\displaystyle a,b \in D$ are associates.
recall that $\displaystyle \delta(x) \leq \delta(xy)$ for all nonzero elements x, y in D. now if $\displaystyle a, b$ are associates,

then $\displaystyle a=ub,$ for some unit $\displaystyle u \in D.$ so $\displaystyle \delta(a) \leq \delta(u^{-1}a) = \delta(b) \leq \delta(ub)=\delta(a). \ \ \square$