1. The problem is to understand the structure of the ring $Q[x,y]$, which is the ring of polynomials in two variables $x$ and $y$ with coefficients in the rational numbers $Q$. 2. The ring $Q[x,y]$ consists of all finite sums of terms of the form $a_{ij}x^iy^j$ where $a_{ij} \in Q$ and $i,j$ are nonnegative integers. 3. Important properties: - $Q[x,y]$ is a commutative ring with unity. - It is an integral domain because $Q$ is a field and polynomial rings over fields are integral domains. - It is not a field because not all polynomials have multiplicative inverses. 4. The ring $Q[x,y]$ can be viewed as $Q[x][y]$ or $Q[y][x]$, i.e., polynomials in $y$ with coefficients in $Q[x]$ or vice versa. 5. This ring is used in algebraic geometry and multivariate polynomial algebra. Final answer: $Q[x,y]$ is the ring of all polynomials in variables $x$ and $y$ with rational coefficients, an integral domain but not a field.