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2014•Unpublished venueRequires access

THE GALOIS GROUP OF THE CHEBYSHEV POLYNOMIALS OF THE FIRST KIND OF PRIME DEGREE

Dele Oluwade, Lokoja Kogi State

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Abstract

The Galois Group of a polynomial p(x) is a group associated with p(x). It is a discrete structure arising from the algebraic Galois Theory of Equations. The Galois Group provides a connection between the algebraic theories of fields and groups. There is a close relationship between the roots of a polynomial and its Galois Group, to wit, the Galois Group of a polynomial refers to a certain permutation group of the roots of the polynomial. In this paper, it is shown that the Galois group of the Chebyshev polynomials of the first kind of prime degree over the field of rationals (a field of zero characteristic) is isomorphic to the cyclic group of order two. The result is established via the concept of a splitting field.

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What this paper is about

The Galois Group of a polynomial p(x) is a group associated with p(x). It is a discrete structure arising from the algebraic Galois Theory of Equations. The Galois Group provides a connection between the algebraic theories of fields and groups. There is a close relationship between the roots of a polynomial and its Galois Group, to wit, the Galois Group of a polynomial refers to a certain permutation group of the roots of the polynomial. In this paper, it is shown that the Galois group of the Chebyshev polynomials of the first kind of prime degree over the field of rationals (a field of zero characteristic) is isomorphic to the cyclic group of order two. The result is established via the concept of a splitting field.

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Available abstract

The Galois Group of a polynomial p(x) is a group associated with p(x). It is a discrete structure arising from the algebraic Galois Theory of Equations. The Galois Group provides a connection between the algebraic theories of fields and groups. There is a close relationship between the roots of a polynomial and its Galois Group, to wit, the Galois Group of a polynomial refers to a certain permutation group of the roots of the polynomial. In this paper, it is shown that the Galois group of the Chebyshev polynomials of the first kind of prime degree over the field of rationals (a field of zero characteristic) is isomorphic to the cyclic group of order two. The result is established via the concept of a splitting field.

Key concepts: Galois group, Mathematics, Embedding problem, Generic polynomial, Fundamental theorem of Galois theory, Galois cohomology, Galois extension, Differential Galois theory

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