Let $D = \mathbb{R} + X \mathbb{C}[X]$
Show that $\gcd_D(X^2,iX^2)=\emptyset $
Here is my plan so far... (and my questions)
Suppose $f \in \gcd_D(X^2,iX^2) $.
How do I show that because X is prime in $\mathbb{C}$ then $X\mid f$ in $\mathbb{C}[X]$ so therefore $if \in D$.
From $fg = X^2$ and $fh = iX^2$ how can I conclude that $ig=h \in D$ and then that $if \in CD_D(X^2,iX^2)$ which will lead me to a contradiction... I believe.
If anyone can help me fill in the blanks that'd be great!
Hint $\ $ If the gcd $\,g$ exists then by definition $\,f\mid x^2,\,ix^2\iff f\mid g.\,$ Hence $\,x^2\nmid ix^2\,\Rightarrow\, x^2\nmid g.\,$ Also $\ x,ix\mid x^2,ix^2\Rightarrow\,x,ix\mid g,\ $ so $\ g = \ cx,\,\ c \in \Bbb C.\,$ But $\,x\mid cx\,\Rightarrow\,c\in \Bbb R,\,$ contra $\,ix\mid cx.$