# proof of finite inseparable extensions of Dedekind domains are Dedekind

Let $R$ be a Dedekind domain^{} with field of fractions^{} $K$ and $L/K$ be a field extension. We suppose that $K$ has characteristic^{} (http://planetmath.org/characteristic) $p>0$ and that there is a $q={p}^{r}$ such that ${x}^{q}\in K$ for all $x\in L$. In particular, this is satisfied if it is a purely inseparable and finite extension^{}.

We show that the integral closure^{} $A$ of $R$ in $L$ is a Dedekind domain.

We cannot apply the same method of proof as for the proof of finite separable extensions of Dedekind domains are Dedekind, because here $A$ does not have to be finitely generated^{} as an $R$-module.

We use the characterization of Dedekind domains as integral domains^{} in which all nonzero ideals are invertible^{} (see proof that a domain is Dedekind if its ideals are invertible).
Note that for any $x\in A$, ${x}^{q}$ is in $K$ and is integral over $R$ so, by integral closure, ${x}^{q}\in R$.

So, let $\U0001d51e$ be a nonzero ideal in $A$, and let $\U0001d51f$ be the ideal of $R$ generated by terms of the form ${a}^{q}$ for $a\in \U0001d51e$,

$$\U0001d51f={({a}^{q}:a\in \U0001d51e)}_{R}.$$ |

Then, as $R$ is a Dedekind domain, there is a fractional ideal^{} ${\U0001d51f}^{-1}$ of $R$ such that $\U0001d51f{\U0001d51f}^{-1}=R$, and write ${\U0001d51f}_{A}^{-1}$ for the fractional ideal of $A$ generated by ${\U0001d51f}^{-1}$. Then,

$$1\in R=\U0001d51f{\U0001d51f}^{-1}\subseteq {\U0001d51e}^{q}{\U0001d51f}_{A}^{-1}.$$ | (1) |

On the other hand, if ${a}_{1},\mathrm{\dots},{a}_{q}\in \U0001d51e$ and $b\in {\U0001d51f}^{-1}$ then

$${({a}_{1}\mathrm{\cdots}{a}_{q}b)}^{q}=({a}_{1}^{q}b)\mathrm{\cdots}({a}_{q}^{q}b)\in R,$$ |

so ${a}_{1}\mathrm{\cdots}{a}_{q}b$ is integral over $R$ and is in $A$. Therefore, ${\U0001d51e}^{q}{\U0001d51f}_{A}^{-1}\subseteq A$. Combining with (1) gives ${\U0001d51e}^{q}{\U0001d51f}_{A}^{-1}=A$, so $\U0001d51e$ is invertible with inverse^{} ${\U0001d51e}^{q-1}{\U0001d51f}_{A}^{-1}$.

Title | proof of finite inseparable extensions of Dedekind domains are Dedekind |
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Canonical name | ProofOfFiniteInseparableExtensionsOfDedekindDomainsAreDedekind |

Date of creation | 2013-03-22 18:35:42 |

Last modified on | 2013-03-22 18:35:42 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 5 |

Author | gel (22282) |

Entry type | Proof |

Classification | msc 13A15 |

Classification | msc 13F05 |