polylib::splitfield

Splitting field of a polynomial

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

Syntax

polylib::splitfield(p)

Description

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.

Examples

Example 1

We adjoin to the rationals:

polylib::splitfield(x^2+1)

Example 2

A call to polylib::splitfield becomes more interesting for polynomials for of degree at least 3:

polylib::splitfield(x^3-2)

Example 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)

Parameters

p

Univariate polynomial over a field or univariate polynomial expression

Return Values

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.

See Also

MuPAD Functions

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