Galois groups of some iterated polynomials over cyclotomic extensions

Let \(\varphi _p(z)=(z-1)^p+2-\zeta _p\), where \(\zeta _p\in \bar{\mathbb {Q}}\) is a primitive pth root of unity. Building on previous work, we show that the nth iterate \(\varphi _p^n(z)\) has Galois group \([C_p]^n\), an iterated wreath product of cyclic groups, whenever p is not a Wieferich prime.     

Galois groups of multivariate Tutte polynomials

The multivariate Tutte polynomial $\hat{Z}_{M}$ of a matroid M is a generalization of the standard two-variable version, obtained by assigning a separate variable v _e to each element e of the ground set E. It encodes the full structure of M. Let v={v _e } _e??E , let K be an arbitrary field, and suppose M is connected. We show that $\hat{Z}_{M}$ is irreducible over K(v), and give three self-contained proofs that the Galois group of $\hat{Z}_{M}$ over K(v) is the symmetric group of degree n, where n is the rank of M. An immediate consequence of this result is that the Galois group of the multivariate Tutte polynomial of any matroid is a direct product of symmetric groups. Finally, we conjecture a similar result for the standard Tutte polynomial of a connected matroid.     

On the Galois groups of Legendre polynomials

Ever since Legendre introduced the polynomials that bear his name in 1785, they have played an important role in analysis, physics and number theory, yet their algebraic properties are not well-understood. Stieltjes conjectured in 1890 how they factor over the rational numbers. In this paper, assuming Stieltjes’ conjecture, we formulate a conjecture about the Galois groups of Legendre polynomials, to the effect that they are “as large as possible,” and give theoretical and computational evidence for it.     

Galois groups of polynomials arising from circulant matrices

Circulant matrices are used to construct polynomials, associated with Chebyshev polynomials of the first kind, whose roots are real and made explicit. Then the Galois groups of the polynomials are computed, giving rise to new examples of polynomials with cyclic Galois groups and Galois groups of order p(p−1) that are generated by a cycle of length p and a cycle of length p−1.     

The Galois groups of the polynomials Xn + aX1 + b

We give conditions under which the Galois group of the polynomial Xⁿ + aX¹ + b over the rational number field Q is isomorphic to the symmetric group S_n of degree n. Using the result, we prove the Williams-Uchiyama conjecture concerning the irreducibility of the polynomial Xⁿ + X + a modulo p.     

Finitely generated subgroups of free profinite groups and of some Galois groups

It is proved that, for a (closed) subgroupH of a free profinite or free prosolvable groupF of rankF>1 such thatH contains a nontrivial composition subgroupN ofF, we have rankF< and [F:H]<.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 95–106, July, 1998.     

Periods of polynomials over a Galois ring

The period of a monic polynomial over an arbitrary Galois ring GR(pe,d) is theoretically determined by using its classical factorization and Galois extensions of rings. For a polynomial f(x) the modulo p remainder of which is a power of an irreducible polynomial over the residue field of the Galois ring, the period of f(x) is characterized by the periods of the irreducible polynomial and an associated polynomial of the form (x-1)m+pg(x). Further results on the periods of such associated polynomials are obtained, in particular, their periods are proved to achieve an upper bound value in most cases. As a consequence, the period of a monic polynomial over GR(pe,d) is equal to pe-1 times the period of its modulo p remainder polynomial with a probability close to 1, and an expression of this probability is given.     

On relations between zeros of Galois polynomials

A conjecture ofH. Kleiman says that over certain fields a Galois equation of degree 3 is uniquely determined by its root polynomials. We prove this conjecture for prime degrees 3 and a somewhat smaller class of fields than Kleiman's. In this situation, the ideal of all relations between zeros of the equation has a basis containing root polynomials only, not the equation itself. Giving a large class of counterexamples of degree 4, we disprove Kleiman's conjecture in general.     

Approximating absolute Galois groups

In this paper we identify a class of profinite groups (totally torsion free groups) that includes all separable Galois groups of fields containing an algebraically closed subfield, and demonstrate that it can be realized as an inverse limit of torsion free virtually finitely generated abelian (tfvfga) profinite groups. We show by examples that the condition is quite restrictive. In particular, semidirect products of torsion free abelian groups are rarely totally torsion free. The result is of importance for K-theoretic applications, since descent problems for tfvfga groups are relatively manageable.     

On vectorial polynomials and coverings in characteristic 3

For a field containing the finite field we give explicitly the whole family of Galois extensions of with Galois group or and determine the discriminant of such an extension.

     

Relatively projective groups as absolute Galois groups

By two well-known results, one of Ax, one of Lubotzky and van den Dries, a profinite group is projective iff it is isomorphic to the absolute Galois group of a pseudo-algebraically closed field. This paper gives an analogous characterization of relatively projective profinite groups as absolute Galois groups of regularly closed fields. Dedicated to Yuri Ershov on the occasion of his 60-th birthday Heisenberg-Stipendiat der Deutschen Forschungsgemeinschaft (KO 1962/1-1).     

On Galois spectra of polynomials with integral parameters

We prove that there exists a polynomial F(x, t) with rational coefficients, whose degree with respect to x is equal to 4, such that for every integer a, the Galois group of the decomposition field of the polynomial F(x, a) is not the dihedral group, but any other transitive subgroup of the group S₄ can be represented as the Galois group of the decomposition field of the polynomial F(x, a) for a certain integer a. Bibliography: 1 title. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 321, 2005, pp. 275–280.     

Small maximal pro-p Galois groups

For an odd primep we classify the pro-p groups of rank ≤4 which are realizable as the maximal pro-p Galois group of a field containing a primitive root of unity of orderp.