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The following exercise and hint appear in Neukirch's Algebraic Number Theory (Section 9, Exercise 3, page 58)

Let $L/K$ be a solvable extension of prime degree $p$ (not necessarily Galois). If the unramified prime ideal $\mathfrak{p}$ in $L$ has two prime factors $\mathfrak{P}$ and $\mathfrak{P'}$ of degree 1, then it is already totally split (theorem of F.K. Schmidt).

Neukirch also gives this hint:

If $G$ is a transitive solvable permutation group of prime degree $p$, then there is no nontrivial permutation $\sigma \in G$ which fixes two distinct letters.

Can we use the conclusion of my former question to solve this one? Thanks!

I would like to know how to use "Let $L/K$ be a solvable extension of prime degree $p$".

P.S. I'm new here and I've asked my friend Roun to ask the "former question" for me.

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    I would like to know how to use "Let L / K be a solvable extension of prime degree p"?2011-05-17
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    (i) If you are going to refer to previous question, you need to link to them, not just say "my former question". (ii) You should write the important information **in the question**, not in comments. In particular, the note that "[you] would like to know how to use" the separability hypothesis should be in the **body of the question**, not in a comment.2011-05-17
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    What is the definition of a solvable non-Galois extension?2011-05-17
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    Your link is broken.2011-05-17
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    @Arturo: Thank you for the advice. I've edited the post.2011-05-17
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    @Arturo: Sorry for that. It is fixed now.2011-05-17
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    @dust: When citing a book, it's good form to give the citation; note also that this is exercise 3 in the book, while the previous post are (1) Exercise 1 and (2) Exercise 4; I doubt that you are expected to use Exercise 4 to solve Exercise 3. Also, there is an extensive hint in the book, why not copy that too?2011-05-17
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    @Alex: it means that the Galois group of this extension is solvable but it is not Galois.2011-05-17
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    Hint from Neukirch: "Use the following result of Galois: if $G$ is a transitive solvable permutation group of prime degree $p$, then there is no nontrivial permutation $\sigma\in G$ which fixes two distinct letters."2011-05-17
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    @Arturo:thank you very much for telling me this,I find some difficulty in using this.....so I try not to type to many words....I am sorry.2011-05-17
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    @dust: Actually a solvable non-Galois separable extension means that Galois group of its Galois closure is solvable.2011-07-06

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