Quotient ring is a domain??

i wanted to ask this ring theory related question My question is, is $\frac {\mathbb{C}[X,Y]}{(X^4+X^3Y+Y^4)}$ a domain or not... i know that for it to be a domain the denominator ideal has to be prime ideal in $\mathbb{C}[X,Y]$ How do i use the fact that the polynomial is homogeneous.. I am a bit new to this topic and would appreciate if somebody would explain this to me.?

• By $C$ do you mean the complex numbers? – Nikolas Wojtalewicz Mar 21 '16 at 4:58
• Yes i mean exactly that – Upstart Mar 21 '16 at 5:03

By fundamental theorem of algebra $x^4+x^3+1$ can be factorized in $\mathbb{C}[x]$ so similarly $x^4+x^3y+y^4$ can be factorized in $\mathbb{C}[x,y]$ so the given ring is not an integral domain.
• Frank we use the homogenity in dividing the polynomial by $y^4$ or $x^4$?? – Upstart Mar 21 '16 at 6:41
• Divide(factor out?) $y^4$ – gamma Mar 21 '16 at 6:42
• Not sure what you mean? suppose $x^4+x^3+1=(x-c_1)(x-c_2)(x-c_3)(x-c_4)$ then $x^4+x^3y+y^4=(x-c_1y)(x-c_2y)(x-c_3y)(x-c_4y)$ right? – gamma Mar 21 '16 at 6:46