On solvable septics

Abstract

The problem of solving the general irreducible polynomial in one variable and expressing its roots in radicals has been proposed ages ago. Since then the case of solving general quadratics, cubics and quartics has been solved and it took a few centuries before Abel and Galois demonstrated the impossibility to solve general equations of degree higher than 4. In two recent papers [D] and [H], Dummit and respectively Hagedorn provide methodologies to solve the general solvable quintic and sextic. In this thesis, we shall adopt the techniques in [D] to give a qualitative description of solving solvable irreducible polynomials of arbitrary prime degree and apply these results to solve for two of the roots of x^7+x^6-12x^5-7x^4+28x^3+14x^2-9x+1=0 which can be used to express cos 2 pi/29 in radicals.References[D] D. S. Dummit, Solving Solvable Quintics, Mathematics of Computation, Volume 57, Issue 195, 387-401, 1991.[H] T. R. Hagedorn, General Formulas for Solving Solvable Sextic Equations, Journal of Algebra, Volume 233, Issue 2, 704-757, 2000.