Square-free polynomial factorization in finite field

The following calculator finds all square factors of a polynomial in the finite field. The square-free factorization is the first step in the polynomial factor decomposition process. You can find more details just below the calculator.

Square free polynomial factoring in finite field

Polynomial

Modulus

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Input polynomial

 

Solution

 

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We already have similar calculator for square free decomposition in the field of rational numbers, but for some edge cases it will not work for a polynomial with coefficients in finite field. A square-free polynomial decomposition algorithm is based on calculation of the greatest common divisor (GCD) of the polynomial and its derivative : gcd(A,A').
The derivative of a non-zero degree polynomial is always not zero in the ring of integers or rational field. But it can become zero in finite field. E.g. the polynomial x⁶+x³+2 has zero derivative in F₃ field, since (6 mod 3)x⁵+(3 mod 3)x² = 0x⁵+0x² = 0. A polynomial term with a degree that's a multiple of a field order becomes zero in the derivative.

Square-free decomposition algorithm in finite field

The square-free factorization algorithm for finite field addresses the zero derivative issue, discussed above. The calculator uses the algorithm, described in wikipedia¹ ( without recursion ):

// a(x) - Input polynomial (must be monic)
// p - field order
m⟵1 
do
       //Greatest common divisor calculation     
        c(x) ⟵ gcd( a(x), a'(x) ) 
        w(x) ⟵ a(x)/c(x)
       i ⟵1 
       loop while w(x) !=1 
           y(x) ⟵ gcd(w(x), c(x));
           q(x) ⟵ w(x) / y(x)
           if q(x)!=1  output ⟵ q(x)^i*m

           w(x) = y(x)
           c(x) = c(x)/y(x)
           i=i+1
       end loop
       if c(x)!=1
           a(x) = c(x)^(1/p)
           m ⟵ p*m
       end if
loop until c(x)!=1