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# 2 Functionality of the package

### Sections

1. Methods for rational polynomials
2. Solving a polynomial by radicals

## 2.1 Methods for rational polynomials

- `IsSolvable(` f`)`
- `IsSolvablePolynomial(` f`)`

  returns `true` if the rational polynomial f has a solvable Galois group and `false` otherwise. It signals an error if there exists an irreducible factor with degree greater than 15.

  For a rational polynomial f

- `SplittingField(` f`)`

  returns the smallest field, constructed with `FieldByPolynomial`, that contains all roots of f.

- `GaloisGroupOnRoots(` f`)`

  calculates the Galois group G of the rationals polynomial f as a permutation group with respect to the ordering of the roots of f given as matrices in G!.roots.

  ## 2.2 Solving a polynomial by radicals

- `RootsOfPolynomialAsRadicals(` f`)`

  computes a solution by radicals for the irreducible, rational polynomial f up to degree 15 if this is possible, e. g. if the Galoisgroup of f is solvable. The result is displayed in form of a dvi-file. Additionally a record is returned which contains the roots of f as a list `roots` of matrices, the Galois group on the roots as component `galgrp` and the splitting field of f in two forms; on the one hand the matrix field `K` generated by the roots and on the other hand an algebraic number field `H` created by the defining polynomial of the matrix field. The record also includes a list `cyclics` of matrices which define the splitting field by gradual, cyclic extensions.

  The computation may last very long and doesn't finish for every example if the degree of f is greater than 7.

- `RootsOfPolynomialAsRadicalsNC(` f`,` display`)`

  does essentially the same as `RootsOfPolynomialAsRadicals` except that you can choose if you want to create a dvi-file and display it by setting the boolean display. The function performs no test whether the polynomial f is irreducible. It also doesn't check at the begining if f is solvable, but can therefore be used for polynomials with arbitrary degree.

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