Content deleted Content added
→See also: −wlink already included in the main article |
→Background: provided example |
||
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 factorising 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 factorise 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]]
| |||