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Chebyshev polynomials and Galois groups of De Moivre polynomials.
- Source :
- International Journal of Number Theory; Sep2022, Vol. 18 Issue 8, p1735-1748, 14p
- Publication Year :
- 2022
-
Abstract
- Let n ≥ 3 be an odd natural number. In 1738, Abraham de Moivre introduced a family of polynomials of degree n with rational coefficients, all of which are solvable. So far, the Galois groups of these polynomials have been investigated only for prime numbers n and under special assumptions. We describe the Galois groups for arbitrary odd numbers n ≥ 3 in the irreducible case, up to few exceptions. In addition, we express all zeros of such a polynomial as rational functions of three zeros, two of which are connected in a certain sense. These results are based on the reduction of an irrational of degree 2n to irrationals of degree ≤ n. Such a reduction was given in a previous paper of the author. Here, however, we present a much simpler approach that is based on properties of Chebyshev polynomials. We also give a simple proof of a result of Filaseta et al. [ABSTRACT FROM AUTHOR]
- Subjects :
- CHEBYSHEV polynomials
IRREDUCIBLE polynomials
POLYNOMIALS
NATURAL numbers
Subjects
Details
- Language :
- English
- ISSN :
- 17930421
- Volume :
- 18
- Issue :
- 8
- Database :
- Complementary Index
- Journal :
- International Journal of Number Theory
- Publication Type :
- Academic Journal
- Accession number :
- 157447205
- Full Text :
- https://doi.org/10.1142/S1793042122500889