As the Galois group is reduced and the coefficient field is extended, the minimal polynomial of V is factored into a lower degree polynomial. The obtained splitting field is a extensions of Q by radicals.Īlong with computing the Galois group, we also compute a minimal polynomial of V where V is a primitive element that extends Q to the splitting field.The above step will be repeated so that the Galois group is reduced to the trivial group, thus we obtain a splitting field of a given polynomial.Starting from Q as the initial coefficient field of the given polynomial, we compute a radical to add to the coefficient field to obtain the extended field (Galois theory's conclusion on correspondance between subnormal groups reduction and normal extension of fileds).If all quotients are primes, then the Galois group is solvable.įor a solvable poynomial, we perform the following computation according to the computed series of normal subgroups of the Galois group: Compute quotients of degree of a normal subgroup in the series and its descent subgroup.Compute a series of normal subgroups of the Galois group.Compute a Galois group of a given polynomial.To determine if a given polynomial is solvable or not, we need to do the followings: Galois theory developed by Évariste Galois says that if a Galois group of a given polynomial is a solvable group, then the polynomial is solvable by radicals. The better description of what you can do with this package is given in the next section. If it is solvable, then to compute the radical solutions based on the Galois Group and Field theory.To determine if a given polynomial is solvable by radicals or not.There are two goals achieved in this package: I use Maxima, a free computer algebra system, for this computation. This is NOT a numerical root finding, but it is truly an algebraic symbolic computation of yielding radical solutions based on coefficients of a given polynomial. However, the same theory developed by Galois also tells us that some higher degree polynomials can be solved with combination of radicals (nth-roots of numbers). That was proven many years ago by the work of mathematicians Abel, Ruffini, and Galois. You may know that higher degree polynomials cannot be solved because there is no formulae for solving polynomials of fifth degree or higher. That is why I started on my blog a series of articles (in Japanese) that describe a Maxima program that performs above all. More surprisingly, there is a very little of such books that mention the possible solvability of higher degree polynomials. Galois group should be computed without solving the equations. The book (and many introductory books in Japan) demonstrates the Galois group concept using the solution formulae of qudratic and cubic equations. A Maxima package for solving polynomials based on their Galois Groups Motivationįew months ago, I read a book about Galois's work on insolvability of fifth degree polynomials.
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