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In algebraic number theory, the Dedekind–Kummer theorem describes how a prime ideal in a Dedekind domain factors over the domain's integral closure. It is named after Richard Dedekind who developed the theorem based on the work of Ernst Kummer.

Statement for number fields

Let ${\displaystyle K}$ be a number field such that ${\displaystyle K=\mathbb {Q} (\alpha )}$ for ${\displaystyle \alpha \in {\mathcal {O}}_{K}}$ and let ${\displaystyle f}$ be the minimal polynomial for ${\displaystyle \alpha }$ over ${\displaystyle \mathbb {Z} }$. For any prime ${\displaystyle p}$ not dividing ${\displaystyle ]}$, write $${\displaystyle f(x)\equiv \pi _{1}(x)^{e_{1}}\cdots \pi _{g}(x)^{e_{g}}\mod p}$$ where ${\displaystyle \pi _{i}(x)}$ are monic irreducible polynomials in ${\displaystyle \mathbb {F} _{p}}$. Then ${\displaystyle (p)=p{\mathcal {O}}_{K}}$ factors into prime ideals as $${\displaystyle (p)={\mathfrak {p}}_{1}^{e_{1}}\cdots {\mathfrak {p}}_{g}^{e_{g}}}$$ such that ${\displaystyle N({\mathfrak {p}}_{i})=p^{\deg \pi _{i}}}$, where ${\displaystyle N}$ is the ideal norm.

Statement for Dedekind domains

The Dedekind-Kummer theorem holds more generally than in the situation of number fields: Let ${\displaystyle {\mathcal {o}}}$ be a Dedekind domain contained in its quotient field ${\displaystyle K}$, ${\displaystyle L/K}$ a finite, separable field extension with ${\displaystyle L=K}$ for a suitable generator ${\displaystyle \theta }$ and ${\displaystyle {\mathcal {O}}}$ the integral closure of ${\displaystyle {\mathcal {o}}}$. The above situation is just a special case as one can choose ${\displaystyle {\mathcal {o}}=\mathbb {Z} ,K=\mathbb {Q} ,{\mathcal {O}}={\mathcal {O}}_{L}}$).

If ${\displaystyle (0)\neq {\mathfrak {p}}\subseteq {\mathcal {o}}}$ is a prime ideal coprime to the conductor ${\displaystyle {\mathfrak {F}}=\{a\in {\mathcal {O}}\mid a{\mathcal {O}}\subseteq {\mathcal {o}}\}}$ (i.e. their sum is ${\displaystyle {\mathcal {O}}}$). Consider the minimal polynomial ${\displaystyle f\in {\mathcal {o}}}$ of ${\displaystyle \theta }$. The polynomial ${\displaystyle {\overline {f}}\in ({\mathcal {o}}/{\mathfrak {p}})}$ has the decomposition $${\displaystyle {\overline {f}}={\overline {f_{1}}}^{e_{1}}\cdots {\overline {f_{r}}}^{e_{r}}}$$ with pairwise distinct irreducible polynomials ${\displaystyle {\overline {f_{i}}}}$. The factorization of ${\displaystyle {\mathfrak {p}}}$ into prime ideals over ${\displaystyle {\mathcal {O}}}$ is then given by $${\displaystyle {\mathfrak {p}}={\mathfrak {P}}_{1}^{e_{1}}\cdots {\mathfrak {P}}_{r}^{e_{r}}}$$ where ${\displaystyle {\mathfrak {P}}_{i}={\mathfrak {p}}{\mathcal {O}}+(f_{i}(\theta ){\mathcal {O}})}$ and the ${\displaystyle f_{i}}$ are the polynomials ${\displaystyle {\overline {f_{i}}}}$ lifted to ${\displaystyle {\mathcal {o}}}$.

References

1. ^ Neukirch, Jürgen (1999). Algebraic number theory. Berlin: Springer. pp. 48–49. ISBN 3-540-65399-6. OCLC 41039802.
2. Conrad, Keith. "FACTORING AFTER DEDEKIND" (PDF).

• Algebraic number theory