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Let $K|F$ be a finite cyclic Galois extension of number fields of degree prime to $p$ with Galois group $H$, where $p$ denotes a rational prime. Let $L|K$ denote a pro-$p$-extension (possibly infinite) with Galois group $G$. Now I denote by $G_1$ the maximal abelian pro-$p$-factor group, e.g. $G_1 = G / Frattini(G)$ with fixed field $L^{ab}$

Now $H$ operates via conjugation on $G$ and even on $G_1$, so $G_1$ becomes a $H$-module of exponent $p$, hence a (semisimple) $\mathbb{F}_p[H]$-module.

My Question is: If I look at the representation $G_1$ of $H$ over the field $\mathbb{F}_p$, what is an equivalent formulation of $G_1$ containing the trivial representation $W$ in terms of Galois theory?

Attempt If $G_1$ contains the trivial representation, it can be written as a direct sum of vector spaces $G_1 = W \oplus V$. feeling is that the group $G_1$ must be a direct product of some subgroups, but I can't figure out how these must look like.

Thank you for your help :)

EDIT I actually want to prove that, if there is an abelian finite $p$-extension of the ground field $M|F$ with Galois group $N$, then $G_1$ must contain the trivial representation. So let's start with this extension. Then we can build the compositum of $k$ and $M$, let's denote it by $M' = Mk$. Then the Galois group of $M'|k$ is canonically isomorphic to $N$ and so $Gal(M'/k)$ is a quotient of $G_1$.

EDIT 2 By translation theorem of Galois theory I now know that $$Gal(M'/F) = N \times H$$ and this means that $H$ operates trivially on $N$, the quotient of $G_1$ by $Gal(L^{ab}/M')$.

How does that translate into representation theory of $G_1$ over $\mathbb{F}_p$?

I know that $N$ is a finite abelian $p$-group that is a quotient of a finite abelian group of exponent $p$. Does the representation of $G_1$ over $\mathbb{F}_p$ has a subrepresentation isomorphic to the trivial representation?

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  • $\begingroup$ Corrected the field theoretic interpretation, for a group-theoretic version see this thread. $\endgroup$ – BIS HD Oct 25 '13 at 14:01
  • $\begingroup$ Galois theory says there is a sequence $H\to{\rm Gal}(L|F)\to G$. How does $H$ act on $G$? $\endgroup$ – whacka Feb 25 '15 at 23:53

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