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We compute the F p -dimension of an n-th graded piece G (n) /G (n+1) of the Zassenhaus filtration for various finitely generated pro-p-groups G. These groups include finitely generated free pro-p-groups, Demushkin pro-p-groups and their free pro-p products. We provide a unifying principle for deriving these dimensions.

Stimulated radiative recombination has been demonstrated experimentally in e − +H + and e − + He + collisions using a merged electron-ion beams apparatus with field ionization detection of the excited neutral products. Enhancement of the recombination over spontaneous recombination to form the n = 11, 12 and 13 levels of atomic hydrogen and the n = 11 and 12 levels of atomic helium by factors of between 1000 and 3000 have been found using a CO 2 laser power of 8 W. Evidence for the resolution of fine-structure levels has been seen for the case of helium.

We observe that some fundamental constructions in Galois theory can be used to obtain some interesting restrictions on the structure of Galois groups of maximal p-extensions of fields containing a primitive pth root of unity. This is an extension of some significant ideas of Demushkin, Labute and Serre from local fields to all fields containing a primitive pth root of unity. Our techniques use certain natural simple Galois extensions together with some considerations in Galois cohomology and Massey products.Theorem 3.8 implies in particular that G cannot be isomorphic to G F (p) for any field F containing a primitive p-th root of unity. The case s = 1 has already been implied by [Ro, Theorem 5.1.2 and Theorem 5.2.1]. Observe that the relations (1) and (2) are quite similar yet they behave differently in Galois theory. We were well acquainted with [Lab, Proposition 6] and its relevance to our work. We realized that it allows a generalization to the infinite case. (See Lemma 3.2.) Throughout our paper a prominent role is played by simple Galois extensions F(a, m) = F( p m √ a, ζ p m ) of F introduced in Section 2. (See also [Ro, Chapter 5], where we introduced these extensions for m = 1 and RELATIONS IN THE MAXIMAL PRO-p QUOTIENTS OF ABSOLUTE GALOIS GROUPS 3 2 in our examples which illustrate our ideas. The general case m ∈ N is an immediate extension of these examples.)As we mentioned above, in [Lab], Labute classified all Demushkin groups and in this way all G F (p), where F is a local field. He provided explicit descriptions of relations in these groups. It is interesting to clarify to what extend we can generalize Labute's result to all fields. Our results form a contribution to this problem. We mentioned some of these ideas to C. Quadrelli in the fall of 2013 and also in later discussions. I. Efrat and C. Quadrelli developed a nice group-theoretical approach to this project. Their paper [EQ] complements well our paper, and we feel that both papers form a tribute to the remarkable thesis of John Labute.We hope that our paper will appeal to a broad audience. In particular this paper should be accessible to graduate students Organization of our paper is as follows. In Section 2 we introduce our basic extensions F(a, m) which we use substantially throughout the paper to show that some relations in G F (p) cannot occur. In Section 3 we recall and slightly generalize parts of the Proposition 6 in [Lab]. We then prove the main results, Theorem 3.5, Theorem 3.8, Theorem 3.10 and Theorem 3.11, which were previously illustrated in [Ro] in a few examples. Section 4 introduces a natural union CR(F) of all F(a, m) for all a ∈ F × and all m ∈ F, called the p-cyclotomic radical extension of F. This section is an extension and continuation of Section 3. In the last section, we consider a type of automatic Galois realization (Theorem 5.7) and use it to provide also some restrictions on the shape of relations in G F (p) (Theorem 5.9).Acknowledgements. First of all we would like to thank Professor John Labute for his remarkable ...

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