# Difference between $F[x,y]$ and $F(x)[y]$ in Ring theory

I come across following ring

$$F[x,y]$$ and $$F(x)[y]$$ where $$F$$ is a field.

I think both are same initially . But as $$xy$$ is irreducible in $$F(x)[y]$$ but reducible in $$F[x,y]$$.

Which is a bigger ring than another?

Any Help will be appreciated.

$$F[x,y]$$ are polynomials with variables $$x,y$$ and coefficients in $$F$$, while $$F(x)[y]$$ are polynomials with variable $$y$$ with coefficients from the field of the rational functions over $$F$$, noted $$F(x)$$.

The difference is that $$F[x,y]=(F[x])[y]$$, meaning that these are polynomials in $$y$$ with coefficients being polynomials in $$F[x]$$, but in $$F(x)[y]$$ the coefficients are not only polynomials in $$F[x]$$, but all rational functions $$p/q$$ with $$p,q \in F[x], q\neq0$$.

We have $$F[X,Y]=F[X][Y]$$, but not $$F[X,Y]=F(X)[Y]$$, because $$F(X)$$ is the fraction field of the integral domain $$F[X]$$, which is "bigger".

$$F [x,y]$$ is the polynomial ring in the variables $$x$$ and $$y$$ with coefficients in $$F$$.

$$F (x)$$ is the quotient field of the polynomial ring $$F [x]$$,that is, $$F (x)$$ contains polynomials of the form $$\dfrac {f (x)}{g (x)}$$, where $$f (x),g (x)\in F [x] ,g (x)\neq 0$$.

Now $$F (x)[y]$$ is the polynomial ring in the variable $$y$$ with coefficients in $$F (x)$$.

Let $$f (y)=\dfrac{y}{x^2} \in F (x)[y]$$. Does $$f (y)$$ belong to $$F [x,y]$$?