Galoisteori for palindrome polynomer

Abstract

In 1862, substituting for professor O. J. Broch, Ludvig Sylow gives the first lectures ever given on the field of Galois theory in Norway, at the University of Christiania. He lectures on, amongst other things, what he calls ``reciproke ligninger'' (reciprocal equations). It turns out that there are some interesting relations between the solutions of these equations. About one hundred and fifty years later, the Norwegian Julie Kjennerud, who majored in mathematics at the University of Oslo in 1938 but later worked as university lecturer in botany, puzzles with some notes she has and tells an old colleague about them. She is then over 100 years old, but have discovered some interesting polynomials, which she calls ``koeffisient symmetriske polynomer'' (coefficient symmetrical polynomials) and which have roots with certain properties. They turn out to be exactly the type of polynomials Sylow lectured about. These polynomials and their roots is exactly the theme of this thesis: we shall look into these special types of polynomials, calling them palindromic polynomials. What can a polynomial's coefficients possibly tell us about its roots? And can certain connections between the roots help us calculate the polynomial's Galois group? We will see how symmetry of the coefficient leads to special pairwise connections between the roots. Then we will use these connections to derive formulas for finding roots of these polynomials up to and including degree nine. Having developed these tools we will consider the Galois groups of the palindromic polynomials, before we ``upper our game'' and consider polynomials which are not precisely palindromic, and which we shall call semipalindromic polynomials. Only some of their roots have the pairwise connection of the palindromic polynomials'. How can we detect that a polynomial is semipalindromic, and what do the Galois groups of the semipalindromic polynomials look like?

In 1862, substituting for professor O. J. Broch, Ludvig Sylow gives the first lectures ever given on the field of Galois theory in Norway, at the University of Christiania. He lectures on, amongst other things, what he calls ``reciproke ligninger'' (reciprocal equations). It turns out that there are some interesting relations between the solutions of these equations. About one hundred and fifty years later, the Norwegian Julie Kjennerud, who majored in mathematics at the University of Oslo in 1938 but later worked as university lecturer in botany, puzzles with some notes she has and tells an old colleague about them. She is then over 100 years old, but have discovered some interesting polynomials, which she calls ``koeffisient symmetriske polynomer'' (coefficient symmetrical polynomials) and which have roots with certain properties. They turn out to be exactly the type of polynomials Sylow lectured about. These polynomials and their roots is exactly the theme of this thesis: we shall look into these special types of polynomials, calling them palindromic polynomials. What can a polynomial's coefficients possibly tell us about its roots? And can certain connections between the roots help us calculate the polynomial's Galois group? We will see how symmetry of the coefficient leads to special pairwise connections between the roots. Then we will use these connections to derive formulas for finding roots of these polynomials up to and including degree nine. Having developed these tools we will consider the Galois groups of the palindromic polynomials, before we ``upper our game'' and consider polynomials which are not precisely palindromic, and which we shall call semipalindromic polynomials. Only some of their roots have the pairwise connection of the palindromic polynomials'. How can we detect that a polynomial is semipalindromic, and what do the Galois groups of the semipalindromic polynomials look like?