## Mild pro-$$p$$-groups and Galois groups of $$p$$-extensions of $$\mathbb Q$$.(English)Zbl 1122.11076

Let $$p$$ be an odd prime and let $$G = F/R$$ be a finitely presented pro-$$p$$-group, where $$F$$ is the free pro-$$p$$-group of generator rank $$m$$ and $$R$$ the closed normal subgroup of $$F$$ with minimal set of generators $$\{r_1,\ldots ,r_d\}.$$ Let $$gr(G) = \bigoplus_{n=1}^{\infty}gr_n(G)$$ be the graded Lie algebra associated to the $$p$$-central series $$\{G_n|n\in \mathbb N\}$$ of $$G$$. For $$r\in F$$ the largest $$n$$ with $$r\in F_n$$ is called the filtration degree of $$r$$ and $$\overline{r}\in gr_n(F)$$ the initial form of $$r$$. Let $$\rho_i$$ be the initial form of $$r_i$$ and let $${\mathfrak r}$$ be the the ideal of $$L:= gr(F)$$ generated by $$\rho_1,\ldots ,\rho_d$$. Then $$L$$ is a Lie algebra over $$\mathbb F_p[\pi]$$, where $$\pi$$ is induced by $$x\to x^p$$.
The sequence $$\rho_1,\ldots ,\rho_d$$ is called strongly free if the enveloping algebra $$U_{{\mathfrak g}}$$ of $${\mathfrak g}:= L/{\mathfrak r}$$ is a free $$\mathbb F_p[\pi]$$-module and $$M:={\mathfrak r}/[{\mathfrak r},{\mathfrak r}]$$ is a free $$U_{{\mathfrak g}}$$-module on the images of $$\rho_1,\ldots ,\rho_d$$ in $$M$$. In this case one says that the presentation $$G=F/R$$ is strongly free. $$G$$ is called mild if it has a strongly free representation. This notion is due to D. Anick [J. Algebra 111, 154–165 (1987; Zbl 0635.20015)]. The author of the paper at hand proves among other results that a mild group has cohomological dimension $$2$$.
Let $$S$$ be a finite set of primes not containing $$p$$ and let $$G_S(p)$$ be the maximal $$p$$-extension of $$\mathbb Q$$ unramified outside $$S$$. Since a prime $$q\in S$$ can be ramified only if $$q\equiv 1 \bmod p$$ we assume $$q\equiv 1 \bmod p$$ for $$q\in S$$. Then $$G_S(p)$$ is a pro-$$p$$-group of generator and relation rank $$|S|$$ [I. R. Shafarevich, Am. Math. Soc., Transl., II. Ser. 59, 128–149 (1966; Zbl 0199.09707); translation from Publ. Math., Inst. Haut. Etud. Sci. 18 (1963), 295–319 (1964; Zbl 0118.27505)]. The structure of the relations is known up to third commutators [H. Koch, J. Reine Angew. Math. 219, 30–61 (1965; Zbl 0137.02402), see as well H. Koch, Galois theory of $$p$$-extensions, Springer (2002; Zbl 1023.11002)]. The structure of $$G_S(p)/G_S(p)^{(3)}$$ was already determined by A. Fröhlich in his paper [“On fields of class two”, Proc. Lond. Math. Soc., III. Ser. 4, 235–256 (1954; Zbl 0055.03301)].
In the paper at hand the author shows that $$G_S(p)$$ is a mild group if the primes of $$S$$ satisfy a certain condition. E.g. this condition is satisfied if $$p=3$$ and $$S=\{7,13,19,31\}$$. Hence in this case $$G_S(3)$$ is a pr-$$3$$-group of cohomological dimension $$2$$ against the earlier belief of many mathematicians that the groups $$G_S(p)$$ never have cohomological dimension 2 (if $$p$$ is not in $$S$$). More general the author proves that for any finite set $$S$$ of primes $$q\equiv 1\bmod p$$ with $$|S|\geq 2$$ there exists an extension of $$S$$ to a set $$S'$$ with $$|S'|= 2|S|$$ such that $$G_{S'}(p)$$ is a mild group.

### MSC:

 11R32 Galois theory 20E18 Limits, profinite groups 11S15 Ramification and extension theory 11S20 Galois theory
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### References:

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