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Let $f(X) \in \mathbb{Z}[X]$ be a monic irreducible polynomial. Let $\theta$ be a root of $f(X)$. Let $A = \mathbb{Z}[\theta]$. Let $K$ be the field of fractions of $A$. Let $B$ be the ring of algebraic integers in $K$. Let $\mathfrak{f}$ be the conductor of $A$, i.e. $\mathfrak{f}$ = {$\alpha \in A; \alpha B \subset A$}.

Is there any relation between the discriminant of $f(X)$ and $\mathfrak{f}$?

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Makoto Kato
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Let $d$ be the discriminant of $f(X)$. Let $\mathfrak{D}_{K/\mathbb{Q}}$ be the different. Then $f'(\theta)B = \mathfrak{f}\mathfrak{D}_{K/\mathbb{Q}}$ (e.g. Neukirch, Ch. III, Th. 2.5). Taking norms of the both sides, we get $|d| = N(\mathfrak{f})|D|$, where $D$ is the discriminant of $K$.

Makoto Kato
  • 42,602