I'm looking at the reducibility of a polynomial $P[X,Y]=X^3 + (Y^4-1)X-(Y^2+1)$ in $\mathbb{R}[Y][X]$, as a polynomial of $X$.
I noticed that this is an Eisenstein polynomial with respect to $Y^2+1$. So $P[X,Y]$ is irreducible in the quotient field of $\mathbb{R}[Y][X]$ if I'm not mistaken.
Now, I'm wondering what is it's quotient field. The book that I'm using used $R(Y)[X]$. Why is that?
Also, I noticed that $P[X,Y]$ is primitive, so $P[X,Y]$ is also irreducible in $\mathbb{R}[Y][X]$. This $\mathbb{R}[Y][X] = \mathbb{R}[X,Y]$. Is this last equality just a matter of notation?
The correct result may be stated as follows : If $R$ is a UFD with quotient field $F$ and $P\in R[X]$ is primitive and irreducible, it is also irreducible in $F[X]$.
Now apply this to $R=F[Y]$ to get irreducibility of $P$ in $\mathbb{R}[X,Y](=\mathbb{R}[Y][X])$ and in $\mathbb{R}(Y)[X]$.
The "equality" $\mathbb{R}[Y][X]=\mathbb{R}(X,Y)$ is obviously false, since the dimension as a real vector space of the l.h.s. is countable while the dimension of r.h.s. is uncountable. Or simply because I don't see how you could write $\dfrac{1}{Y}$ as a polynomial in $X$ and $Y$...