# Problem with the hyperelliptic equation

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.

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.