## A class-field theoretical calculation.(English)Zbl 1161.11402

Let $$k$$ be a function field of one variable over a finite field $$\mathbb{F}_p$$, where $$p$$ is a prime number. Let $$K/k$$ be a finite abelian extension with Galois group $$\Gamma:=\text{Gal}(K/k)$$. Denote by $$S$$ the set of ramified primes of $$K$$ in $$K$$ over $$k$$. Let $$K_S^{ab,p}$$ be the maximal pro-$$p$$ abelian extension of $$K$$ unramified outside $$S$$ with Galois group $$H:=\text{Gal}(K_S^{ab,p}/K)$$. Then $$K_S^{ab,p}$$ is Galois over $$k$$ with Galois group $$G:=\text{Gal}(K_S^{ab,p}/k)$$. Thus there is an exact sequence $1\to H\to G\to \Gamma \to 1.$ The action of $$\Gamma$$ on $$H$$ is $$\gamma\circ h:=\tilde{\gamma}h\tilde{\gamma}^{-1},$$ for all $$h\in H$$ and $$\gamma\in \Gamma$$, where $$\tilde{\gamma}$$ is any lift of $$\gamma$$ to $$G$$. Consider the augmentation ideal $$I_{\Gamma}:=\langle\gamma-1|\gamma\in\Gamma\rangle$$ and $$H$$, we have $$(\gamma-1)\circ h=\tilde{\gamma}h\tilde{\gamma}^{-1}h^{-1}=$$ the commutator $$[\tilde{\gamma},h]$$ of $$\tilde{\gamma}$$ and $$h$$, for all $$\gamma\in\Gamma$$ and $$h\in H$$. As a result, we have an the inclusion $$I_{\Gamma}\circ H\subseteq [G,G]$$, where $$[G,G]$$ is the commutator subgroup of $$G$$. This paper gives a necessary and sufficient condition for the other inclusion $$I_{\Gamma}\circ H \supseteq [G,G]$$ to hold, i.e. $$I_{\Gamma}\circ H =[G,G]$$ if and only if the $$p$$-sylow subgroup $$\Gamma^{(p)}$$ of $$\Gamma$$ is cyclic.
The author first gives a sufficient condition for $$I_{\Gamma}\circ H =[G,G]$$ to hold, i.e. the condition $$H/I_{\Gamma}\circ H$$ has no torsion; then using class field theory he shows an isomorphism $$T(H/I_{\Gamma}\circ H)\cong \wedge^2 \Gamma^{(p)}$$, where $$T(H/I_{\Gamma}\circ H)$$ is the torsion subgroup of $$H/I_{\Gamma}\circ H$$; thus he concludes that if $$\Gamma^{(p)}$$ is cyclic then $$I_{\Gamma}\circ H =[G,G]$$; finally the author uses class field theory and a theorem of H. Kisilevsky [J. Number Theory 44, No. 3, 352–355 (1993; Zbl 0780.11058)] to show that the condition $$\Gamma^{(p)}$$ is cyclic is in fact necessary.

### MSC:

 11R37 Class field theory 11R32 Galois theory

### Keywords:

class field theory; Galois cohomology

Zbl 0780.11058
Full Text:

### References:

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