Prove that polynomial ring $R[X][Y]$ is the same as ring of polynomials in $2$ variables $R[X,Y]$.

Question

Prove that the polynomial ring $R[X][Y]$ is the same as ring of polynomials in 2 variables: $R[X,Y]$.

Unfortunately I do not have intuition about first ring, so I do not see how to establish a bijection between the rings.

Answer 1

Your first ring is polynomials in $Y$ where the coefficients are (polynomials in $X$ with coefficients in $R$). For example, if $R = \Bbb Z$, $(3X + 7X^2)Y^2 + (2 + X)Y + (27X^3 + 4X + 1)\in\left(\Bbb Z[X]\right)[Y]$. By distributing, you can interpret this as an element of $\Bbb Z[X,Y]$ in a natural way, so you have a map $\phi: \left(\Bbb Z[X]\right)[Y]\to\Bbb Z[X,Y]$. You should check that $\phi$ is a ring homomorphism, and then that it has an inverse.

Hint: Showing that $\phi$ is a ring homomorphism is essentially automatic, to find $\phi^{-1}$ you need to find a way to write a general polynomial in two variables as a polynomial in one of the variables with coefficients in (polynomials in the other variable). Try using $X^2Y^2 + 3X^2Y + 5XY^2 + 2XY + 1\in\Bbb Z[X,Y]$ as an example; write this as an element of $\left(\Bbb Z[X]\right)[Y]$. (Then the general procedure is the same as in this specific case, so you'd just need to generalize the argument to higher degrees and rings $R$ that aren't necessarily $\Bbb Z$, and show that $\phi^{-1}$ is also a ring homomorphism.)