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Given a prime ideal in a Dedekind domain, can we always find a separable extension in which the prime ideal splits? If the answer is no in general, is it true under mild conditions on the Dedekind domain?

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    @Qil: These are the definitions I was using (page 1) http://www.maths.bris.ac.uk/~malab/PDFs/Algae_are_more_numb_9.pdf. In other words, r>1. I thought in a trivial extension every prime ideal would be nonsplit. Am I missing something?2012-10-07

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Let $k$ be the residue field at a given maximal ideal $\mathfrak p$ of a Dedekind domain $A$. Suppose $\mathrm{char}(k)\ne 2$. Let $a\in 1+\mathfrak p$ such that $a$ is not a square in $A$. Consider $ B=A[T]/(T^2-a).$ This is an integral domain finite over $A$. Let $C$ be the integral closure of $A$ in $\mathrm{Frac}(A)$. Then $B\subseteq C$. Let $f=2a$. Then $B_f=A_f[T]/(T^2-a)$ is unramified over $A_f$ because for all $\mathfrak q$ not containing $f$, we have $ B\otimes_A A/\mathfrak q=k(\mathfrak q)[T]/(T^2-\bar{a})$ is reduced. This implies that $B_f$ is integrally closed, hence $B_f=C_f$. Above $\mathfrak p$, $ C\otimes_A A/\mathfrak p=B\otimes_A A/\mathfrak p=k[T]/(T^2-1)$ is direct sum of two copies of $k$, so there at least (hence exactly) two prime ideals of $C$ above $\mathfrak p$. The extension is actually completely split.

If $\mathrm{char}(k)=2$, one can use one equation of the form $T^2+T+a$.

Remark An element $a$ as at the begininning usually exist: if $\mathfrak q$ be another maximal ideal of $A$, use Chine Remainder Theorem to find $a\equiv 1 \mod \mathfrak p, \quad a\equiv 0 \mod \mathfrak q, \quad a\not\equiv 0 \mod \mathfrak q^2.$ If $A$ is a local ring with uniformizing element $\pi$, then use the equation $(T-1)^2T^2-\pi=0$.

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    @Khan: now I think it is not necessary to prove that $C$ coincides with $B$ over $A_f$. We see that there are two maximal ideals of $B$ lying over $\mathfrak p$. As $C$ is finite over $B$, each of these ideals lifts to a maximal ideal of $C$. So there are at least two maximal ideals of $C$ lying over $\mathfrak p$.2012-10-07