Splitting field of a polynomial
This functionality does not run in MATLAB.
Given a polynomial p over a field K in one indeterminate X, polylib::splitfield(p) returns a simple field extension F of K and some elements α1, …, αn of F, such that is an associate of p, and such that F is the smallest extension of K containing all of the αi.
If the input is a polynomial expression, as in Example 1, it is treated as a polynomial over the rationals.
The polynomial p need not be irreducible.
The name for the primitive element of the field extension is generated using genident and is therefore different in every call of polylib::splitfield, even if the same polynomial is passed.
MuPAD® must be able to factor polynomials over the coefficient field of p.
The coefficient field must be perfect. Otherwise, it may happen that polylib::splitfield does not terminate.
We adjoin to the rationals:
A call to polylib::splitfield becomes more interesting for polynomials for of degree at least 3:
In this example, we work over the field of univariate rational functions (the quotient field of the univariate polynomials) over the rationals:
R:=Dom::DistributedPolynomial([x], Dom::Rational): F:=Dom::Fraction(R): f:=poly(y^3-x,[y],F): polylib::splitfield(f)
polylib::splitfield returns a list of two operands: the first one is the splitting field of the polynomial, i.e. a Dom::AlgebraicExtension of the coefficient ring; the second one is a list of all roots of the polynomial in the splitting field, each root followed by its multiplicity.