Let $L/K$ be a finite extension of number fields of degree $n$ with $n$ an even integer such that the normal closure of $L$ has the Galois group isomorphic to $D_n$, the dihedral group of order $2n$. Is there any result about the decomposition form of a prime $\mathfrak{p}$ of $K$ in $L$?
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$\begingroup$ Can you be a little more specific as to what you're looking for? Do you understand the general situation for relating splitting in $L/K$ to splitting in the Galois closure? If not, I'd say this is more appropriate for MathStackExchange. $\endgroup$– KimballCommented Jun 15, 2020 at 16:15
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$\begingroup$ Many thanks for your response. I'm looking for an explicit decomposition form in "even dihedral extensions L/K" (or in the Galois closure of L over K) as in Cohen's book (Advanced Topics in Computational Number Theory, Proposition 10.1.26) the decomposition form of primes in a "prime order" dihedral extensions is formulated. $\endgroup$– A. MaarefparvarCommented Jun 16, 2020 at 8:51
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