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.


11R32 Galois theory
20E18 Limits, profinite groups
11S15 Ramification and extension theory
11S20 Galois theory
Full Text: DOI


[1] DOI: 10.1016/0021-8693(82)90104-1 · Zbl 0502.16002
[2] DOI: 10.1016/0021-8693(87)90246-8 · Zbl 0635.20015
[3] DOI: 10.1016/0021-8693(66)90034-2 · Zbl 0146.04702
[4] Halperin S., Math. Scand. 61 pp 1– (1987)
[5] DOI: 10.1002/mana.19690420413 · Zbl 0191.33702
[6] DOI: 10.1002/mana.19770780125 · Zbl 0383.12011
[7] Kuzmin L. V., Izv. Akad. Nauk SSSR 33 pp 1149– (1969)
[8] DOI: 10.1007/BF01425247 · Zbl 0212.36303
[9] DOI: 10.2307/2000425 · Zbl 0576.20022
[10] DOI: 10.2307/2048123 · Zbl 0696.20035
[11] DOI: 10.2307/2006956 · Zbl 0541.20020
[12] Morishita M., Math. 550 pp 141– (2002)
[13] DOI: 10.1023/A:1020038431624 · Zbl 1021.11029
[14] Vogel D., Math. 581 pp 117– (2005)
[15] Waldspurger J.-L., Bull. Soc. Math. 100 pp 113– (1976)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.