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Math Help - generators for one algebra

  1. #1
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    generators for one algebra

    Hello,

    Let I=<x^{2}-\beta y^{2}> 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}.

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by Biscaim View Post
    Hello,

    Let I=<x^{2}-\beta y^{2}> 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}.

    Thanks in advance.
    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]}{<x^2-\beta y>} \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].
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