# Square-free polynomial factorization in finite field

The calculator finds all square factors of polynomial in finite field.

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Creado: 2019-08-26 18:22:02, Última actualización: 2020-11-03 14:19:37

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

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