Galois groups for one class of equations

We find recurrent formulas for obtaining minimal polynomials p n(x) ∈ Z[x] of numbers of the form cos pi/n, where n ∈ N. We demonstrate that Galois groups of these polynomials are commutative. By the same token we give examples of equations of arbitrarily high degrees solvable in radicals. © 2011 World Scientific Publishing Company

One class of equations solvable in radicals

We derive recurrent formulas for obtaining minimal polynomials for values of tangents and show that Galois groups of these polynomials are commutative. Thus we give examples of equations of arbitrarily high degrees solvable in radicals. © 2011 Allerton Press, Inc

Finding minimal polynomials of algebraic numbers of the form tan²(π/n) using Tschirnhausen’s transform

© 2016, Pleiades Publishing, Ltd.Solutions of two problems are proposed based on the Tschirnhausen transform. The first problem is connected with the construction of minimal polynomials of the numbers of the form tan2(π/n) by means of the Tschirnhausen transform for all natural n > 2. The second problem consists in finding the exact roots of the equation x3 − 7x − 7 = 0. A solution of the problem is obtained from the fact that the roots of the equation produce the cyclotomic field Q7. Examples of construction of minimal polynomials are provided

Mathematic simulation of high-efficiency process of grain cleaning

© Published under licence by IOP Publishing Ltd. The article presents the results of field experiment and the results of computer simulation of the grain cleaning process pneumosorting machine PSM-0,5. The results of the comparison of two methods of research are presented

Recurrent formulas for explicitly finding some minimal polynomials

© Published under licence by IOP Publishing Ltd. When solving many applied problems, it becomes necessary to solve nonlinear equations, including the need to find the roots of polynomials. Therefore, the study of the properties of polynomials is an important problem. In the present paper, we obtain recurrence formulas for minimal polynomials over circular fields

Galois groups for one class of equations

We find recurrent formulas for obtaining minimal polynomials p n(x) ∈ Z[x] of numbers of the form cos pi/n, where n ∈ N. We demonstrate that Galois groups of these polynomials are commutative. By the same token we give examples of equations of arbitrarily high degrees solvable in radicals. © 2011 World Scientific Publishing Company

Galois groups for one class of equations

We find recurrent formulas for obtaining minimal polynomials p n(x) ∈ Z[x] of numbers of the form cos pi/n, where n ∈ N. We demonstrate that Galois groups of these polynomials are commutative. By the same token we give examples of equations of arbitrarily high degrees solvable in radicals. © 2011 World Scientific Publishing Company

Normal Basis of the Maximal Real Subfield of a Circular Field

© 2019, Pleiades Publishing, Ltd. We find a necessary and sufficient condition on the natural number n in order that the conjugates of an entire algebraic number α = 2 cos(π/n) form a normal basis of the field ℚ(α); we show that this normal basis at the same time is fundamental

Normal Basis of the Maximal Real Subfield of a Circular Field

© 2019, Pleiades Publishing, Ltd. We find a necessary and sufficient condition on the natural number n in order that the conjugates of an entire algebraic number α = 2 cos(π/n) form a normal basis of the field ℚ(α); we show that this normal basis at the same time is fundamental

One class of equations solvable in radicals

We derive recurrent formulas for obtaining minimal polynomials for values of tangents and show that Galois groups of these polynomials are commutative. Thus we give examples of equations of arbitrarily high degrees solvable in radicals. © 2011 Allerton Press, Inc