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Let $\alpha$ be a root of $f(X)=X^3-X+1$, $K=\mathbb{Q}(\alpha)$ and $L$ the splitting field of $f(X)$, so $Gal(L/\mathbb{Q})=S_3$. This is an old oral exam question and I'm trying to figure out how to determine the densities of different splittings in $K$.

We have the following options for primes $p\in\mathbb{Z}$ that are unramified in $L$:

  1. $(p)=\mathfrak{p}$ in $K\Rightarrow (p)=\mathfrak{P}_1\mathfrak{P}_2$ in $L$.
  2. $(p)=\mathfrak{p}_1\mathfrak{p}_2$ in $K\Rightarrow (p)=\mathfrak{P}_1\mathfrak{P}_2\mathfrak{P}_3$ in $L$.
  3. $(p)=\mathfrak{p}_1\mathfrak{p}_2\mathfrak{p}_3$ in $K\Rightarrow (p)=\mathfrak{P}_1\mathfrak{P}_2\mathfrak{P}_3$ or $(p)$ is totally split in $L$

The problem here is that (2) and (3) share a common type of splitting. Given a prime $\mathfrak{P}$ in $L$, $(p)=\mathfrak{P}\cap\mathbb{Z}$ and such that $(p)=\mathfrak{P}_1\mathfrak{P}_2\mathfrak{P}_3$ in $L$, I would need to know which come from a splitting of type (2) and which from type (3) in order to compute the densities in $K$.

We know that half the primes split going from $K$ to $L$, so it would seem intuitive that exactly half the primes in (3) would split completely in $L$. If this is true, then that would allow us to compute the densities in $K$. However, I don't see a way to prove this as the primes that do split going from $K$ to $L$ might be the primes appearing in the factorizations in (1) and (2).

I'm also curious if there's a way to find out precisely which primes split in which way in $K$? This was a follow-up question, but $L/\mathbb{Q}$ is not abelian and $K/\mathbb{Q}$ is not Galois, so this seems very hard unless there's some trick around it.

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You never quite stated the question. I think you want to know about the splitting of rational primes in $K$, yes? To know what proportion of them split which way? But don't you know how $(p)$ splits in $K$ if you know how it splits in $L$? – Tom Goodwillie Sep 25 '11 at 1:46
There were two questions: 1. Compute the density of each splitting of rational primes in $K$. 2. Find out which primes split how. For the second question I guess an answer would be to write down some congruence relations for the primes in $\mathbb{Q}$ that would determine how they split. However, the extension is not abelian, so I don't have a clue if this can be done. – Edvard F Sep 25 '11 at 15:43

A classical theorem due to Dedekind gives you the prime decomposition of a nonramified prime in any subfield of a normal extension. You can find it e.g. in Marcus' Number Fields.

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Specifically, here is the statement: Let $\mathrm{Gal}(L/\mathbb{Q}) = G$ and let $K$ be the fixed field of $H \subseteq G$. Let $X = G/H$. Let $p$ be a prime which is unramified in $L$. Let $\phi \in G$ be the Frobenius at $p$ (defined up to conjugacy). Then the primes of $K$ lying above $p$ are in bijection with the orbits of $\phi$ acting on $X$, and the degree of the prime is the size of the orbit. – David Speyer Sep 25 '11 at 14:16

The first possibility in your Case 3 never occurs(!):

If $(p)$ splits completely in $K$ it does so also in the other degree $3$ subfields of $L$ (since they are conjugate). Since $L$ is the compositum of any two such subfields it follows that $(p)$ splits completely in $L$ as well.

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Thanks. This is most probably the shortest way to solve the problem. It never occurred to me to look at the conjugates. – Edvard F Sep 25 '11 at 15:39

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