# Math Help - Fews rings question

1. ## Fews rings question

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

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

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

(c)If $\delta$ is Euclidean norm on Euclidean domain $D$ then $\delta(a)=\delta(b)$ if $a,b \in D$ 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 $F$ be field. Every ideal in $F[x]$ is a prime ideal.
this is very false. for example, $$ is not prime.

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

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

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