Content deleted Content added
some crude solecisms corrected |
Adumbrativus (talk | contribs) →Background: Linkify squarefree polynomial |
||
Line 8:
===Background===
The Cantor–Zassenhaus algorithm takes as input a [[squarefree polynomial]] <math>f(x)</math> (i.e. one with no repeated factors) of degree ''n'' with coefficients in a finite field <math>\mathbb{F}_q</math> whose [[irreducible polynomial]] factors are all of equal degree (algorithms exist for efficiently factoring arbitrary polynomials into a product of polynomials satisfying these conditions, for instance, <math>f(x)/\gcd(f(x),f'(x))</math> is a squarefree polynomial with the same factors as <math>f(x)</math>, so that the Cantor–Zassenhaus algorithm can be used to factor arbitrary polynomials). It gives as output a polynomial <math>g(x)</math> with coefficients in the same field such that <math>g(x)</math> divides <math>f(x)</math>. The algorithm may then be applied recursively to these and subsequent divisors, until we find the decomposition of <math>f(x)</math> into powers of irreducible polynomials (recalling that the [[ring (mathematics)|ring]] of polynomials over any field is a [[unique factorisation ___domain]]).
All possible factors of <math>f(x)</math> are contained within the [[factor ring]]
| |||