This is something I started thinking about based on the answer to one of my previous questions. Assume that we have some tower of finite extensions of number fields $L/F/K$ s.t. $L/K$ is Galois. If $\mathfrak{P}$ is a prime in $L$ and $G(\mathfrak{P})$ the decomposition group over $K$ and $I(\mathfrak{P})$ the inertia group over $K$, then what information do we get from the intersections:

$$G(\mathfrak{P})\cap \textrm{Gal}(L/F),\quad I(\mathfrak{P})\cap \textrm{Gal}(L/F)$$

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    $\begingroup$ These will be the decomposition and inertia groups of $\mathfrak{P}$ over $F$. $\endgroup$ – Kevin Ventullo Nov 3 '11 at 18:59

The group $G(\mathfrak{P})$ is the collection of all $\sigma\in\mathrm{Gal}(L/K)$ such that $\sigma(\mathfrak{P}) = \mathfrak{P}$. So $G(\mathfrak{P})\cap\mathrm{Gal}(L/F)$ is the collection of all $\sigma\in\mathrm{Gal}(L/K)$ that fix $F$ pointwise and map $\mathfrak{P}$ to $\mathfrak{P}$ (as a set); i.e., the decomposition group of $\mathfrak{P}$ over $F$.

Likewise, the inertia group $I(\mathfrak{P})$ is the subgroup of all $\sigma\in\mathrm{Gal}(L/K)$ such that $\sigma(a)\equiv a \pmod{\mathfrak{P}}$ for all $a\in\mathcal{O}_L$. Intersecting with $\mathrm{Gal}(L/F)$ gives you the inertia group over $F$.

So, for example, the size of $G(\mathfrak{P})\cap\mathrm{Gal}(L/F)$ gives you the inertial degree of $\mathfrak{P}$ over $\mathfrak{P}\cap\mathcal{O}_F$ (and hence the inertial degree of $\mathfrak{P}\cap\mathcal{O}_F$ over $\mathfrak{P}\cap \mathcal{O}_K$); and the size of $I(\mathfrak{P})\cap\mathrm{Gal}(L/F)$ gives you the ramification index of $\mathfrak{P}$ over $\mathfrak{P}\cap\mathcal{O}_F$. Basically, any information you can generally get out of the decomposition and inertia groups, plus what information you can get out of the multiplicativity of the inertial degrees and ramification indexes in towers.

  • $\begingroup$ Great. The example shows that this is probably the "right" way to compute some of these especially when one knows Frobenius elements etc. Thanks. $\endgroup$ – pki Nov 6 '11 at 0:40

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