If $K$ is a field is $K[X^2,X^3]$ a UFD when considered as a subring of $K[X]$?
I know that $K[X]$ is a $PID$ but nothing else ,these ring theory question just dont seem to occur to me..please help...someone told me it is not a $UFD$ and only just a Domain...but i dont know the reason..how should one go about proving these results?..this ring theory course is getting to hard for my understanding..
On
You might know that UFD's are integrally closed in their field of fractions. I.e. suppose $A$ is a UFD with field of fractions $K$. If an element $a\in K$ satisfies a monic polynomial with entries in $A$, then in fact $a\in A$.
But in your case the element $x$ which is in the field of fractions of $k[x^2,x^3]$. (Check this. It should be easy). But $x$ satisfies the monic polynomial $Y^2-x^2$ which has coefficients in $A$ but $x$ itself is not an element of $A$. So $A$ is not a UFD.
No, $x^2$ and $x^3$ are non-associate irreducibles in the given ring and $x^8=(x^2)^4=(x^3)^2(x^2)$.