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Suppose $K$ is an algebraic number field with $ [ K : \mathbb{Q} ] = d $.

$X, Y , \alpha_1 , \ldots \alpha_n $ are in $O_K$ , i.e. are integral over $\mathbb{Z} $. Suppose that we have the following equation $$Y^2 = (X-\alpha_1) \cdots ( X -\alpha_n) $$ Thus this implies the ideal equation $$[Y^2] = [X-\alpha_1] \cdots [ X -\alpha_n]$$ The book I'm reading ( Baker, Transcendental number theory ) says that we have $$[X - \alpha_j] = \mathfrak{a}\mathfrak{b}^2 $$ where $\mathfrak{a}$ and $\mathfrak{b}$ are ideals in $O_K$ and $\mathfrak{a}$ divides $$ \prod_{i\neq j}[\alpha_j - \alpha_i] $$ But why this is true ? I don't understand this point.

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Start with the simpler case of two coprime factors: If $a$ and $b$ are relatively prime ideals and $ab$ is the square of an ideal, what can you say about the prime factors of $a$ and $b$?

From this, what can we say if $a$ and $b$ are not coprime? Note that if a prime divides the ideals $(X-\alpha_i)$ and $(X-\alpha_j)$, then it divides $(\alpha_j-\alpha_i)$ as well.

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