factorization of primitive polynomial
As an application of the parent entry (http://planetmath.org/EliminationOfUnknown) we take the factorization of a primitive polynomial of into primitive (http://planetmath.org/PrimitivePolynomial) prime factors. We shall see that the procedure may be done by performing a finite number of tests.
be a primitive polynomial in .
If has a primitive quadratic factor, then it has also a factor
It gives finally the remainder where and belong to . According to the parent entry (http://planetmath.org/EliminationOfUnknown) we bring the system
to the form
and then can determine the possible rational solutions of the system via a finite number of tests. Hence we find the possible quadratic factors (1) having rational coefficients. Such a factor is converted into a primitive one when it is multiplied by the gcd of the denominators of and .
Determining a possible cubic factor with rational coefficients requires examination of a remainder of the form
In the needed system
we have to perform two eliminations. Then we can act as above and find a primitive cubic factor of . Similarly also the primitive factors of higher degree. All in all, one needs only look for factors of degree .
- 1 K. Väisälä: Lukuteorian ja korkeamman algebran alkeet. Tiedekirjasto No. 17. Kustannusosakeyhtiö Otava, Helsinki (1950).
|Title||factorization of primitive polynomial|
|Date of creation||2013-03-22 19:20:30|
|Last modified on||2013-03-22 19:20:30|
|Last modified by||pahio (2872)|
|Synonym||primitive factors of primitive polynomial|