# Math Help - generators for one algebra

1. ## generators for one algebra

Hello,

Let $I=$ be an ideal of $\mathbb{Q}[x,y]$ where $\beta\neq a^{2}$ for any $a\in \mathbb{Q}$. Find two generators for the algebra $\frac{\mathbb{Q}[x,y]}{I}$.

Let $I=$ be an ideal of $\mathbb{Q}[x,y]$ where $\beta\neq a^{2}$ for any $a\in \mathbb{Q}$. Find two generators for the algebra $\frac{\mathbb{Q}[x,y]}{I}$.
since $\sqrt{\beta} \notin \mathbb{Q},$ the polynomial $x^2 - \beta y^2$ is irreducible over $\mathbb{Q}[y][x]=\mathbb{Q}[x,y].$ thus: $\frac{\mathbb{Q}[x,y]}{I} = \frac{\mathbb{Q}[y][x]}{} \cong \mathbb{Q}[y][\sqrt{\beta}y]=\mathbb{Q}[y,\sqrt{\beta}y],$ where $\sqrt{\beta} \in \mathbb{C}-\mathbb{Q}.$
note that if $\beta = a^2,$ for some $0 \neq a \in \mathbb{Q},$ then $x^2-\beta y^2=(x- ay)(x+ay)$ and thus by the Chinese remainder theorem: $\frac{\mathbb{Q}[x,y]}{I} \cong \mathbb{Q}[y] \times \mathbb{Q}[y].$