Previous Page Next Page Contents

polylib::splitfield -- the splitting field of a polynomial

Introduction

Given p in K[X], polylib::splitfield(p) returns a simple field extension F of K and some elements alpha1, ...,alphan of F, such that product(X-alphai, i=1..n) is an associate of p, and such that F is the smallest extension of K containing all of the alphai.

Call(s)

polylib::splitfield(p)

Parameters

p - univariate polynomial over a field or univariate polynomial expression

Returns

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.

Related Functions

factor, evalp

Details

Example 1

We adjoin sqrt(-1) to the rationals:

>> polylib::splitfield(x^2+1)
   
                                                2
      [Dom::AlgebraicExtension(Dom::Rational, X1  + 1 = 0, X1),
      
         [X1, 1, -X1, 1]]
      

Example 2

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

>> polylib::splitfield(x^3-2)
   
      --
      |                                           6
      |  Dom::AlgebraicExtension(Dom::Rational, X4  + 108 = 0, X4),
      --
      
         --        4       4              4    -- --
         |  X4   X4      X4        X4   X4      |  |
         |  -- - ---, 1, ---, 1, - -- - ---, 1  |  |
         -- 2    36      18        2    36     -- --

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)
      --
      |
      |  Dom::AlgebraicExtension(Dom::Fraction(
      --
      
         Dom::DistributedPolynomial([x], Dom::Rational, LexOrder)),
      
                               --        4        4              4
             2     6           |  X6   X6       X6        X6   X6
         27 x  + X6  = 0, X6), |  -- - ----, 1, ---, 1, - -- - ----,
                               -- 2    18 x     9 x       2    18 x
      
           -- --
            |  |
         1  |  |
           -- --

Changes




Do you have questions or comments?


Copyright © SciFace Software GmbH & Co. KG 2000