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## On the geometry of Galois cubic fields

The action is considered of a group of totally positive units of a real cubic Galois field on the border of convex hull of its totally positive integers. In case of so called regular fields the fundamental domain of this action has a simple description.

The action of the group of totally positive units on the convex hull of the semigroup of totally positive integers of a real cubic Galois field is studied. The fundamental domain of this action has a simple description in the case of so called regular field.

The article shows the importance of philosophy Ricker for theoretical sociology. Perspectives of sociology associated with a combination of theories and theories of action events. Action theory developed in sociology and theory of events is not. Ricoeur philosophy - one of the possible intellectual resources in order to change this situation.

*In this paper, I will discuss the existing candidates for action-defining entities and structures (the entities and structures which make some X an action) and propose one more candidate. First, I will examine the standard causal theory which became mainstream in analytical philosophy (although this situation is starting to change). Then I will sketch some arguments against the causal theory of action stemming from the works of earlier analytic philosophers, especially from Wittgenstein’s reflections on the nature of the action. Next, I will try to address the problems of action theory by introducing the concept of will as distinguishing feature of actions. Finally, I will discuss the difficulties concerning the concept of will as I construe it in this paper.*

Max Weber. Basic concepts of sociology. Unabridged translation.

The paper outlines key concepts of Hanna Arendt’s political philosophy. The main purpose of the work is to analyze the political virtues — courage, pride and respect, as well as the fundamental abilities (powers) to forgive and to promise. Besides, it’s important to pay attention to Arendt’s understanding of political sphere (as the Web of Relationships) and the role of language in political life. Taking these into account, the main political virtues and abilities of ζῷον πολιτικόν bind together past, present and future of political body into one space of history (ἱστορία).

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.