Factorization : factor

### 5.13.10 Factorization : factor

factor takes as argument an expression.
factor factorizes this expression on the field of its coefficients, with the addition of i in complex mode. If sqrt is enabled in the Cas configuration, polynomials of order 2 are factorized in complex mode or in real mode if the discriminant is positive.
Examples

1. Factorize x4−1 over ℚ.
   Input :
   factor(x`^`4-1)
   Output :
   (x`^`2+1)*(x+1)*(x-1)
   The coefficients are rationals, hence the factors are polynomials with rationals coefficients.
2. Factorize x4−1 over ℚ[i]
   To have a complex factorization, check complex in the cas configuration (red button displaying the status line).
   Input :
   factor(x`^`4-1)
   Output :
   -i*(-x+-i)*(i*x+1)*(-x+1)*(x+1)
3. Factorize x4+1 over ℚ
   Input :
   factor(x`^`4+1)
   Output :
   x`^`4+1
   Indeed x4+1 has no factor with rational coefficients.
4. Factorize x4+1 over ℚ[i]
   Check complex in the cas configuration (red button rouge displaying the status line).
   Input :
   factor(x`^`4-1)
   Output :
   (x`^`2+i)*(x`^`2+-i)
5. Factorize x4+1 over ℝ.
   You have to provide the square root required for extending the rationals. In order to do that with the help of Xcas, first check complex in the cas configuration and input :
   solve(x`^`4+1,x)
   Output :
   [sqrt(2)/2+(i)*sqrt(2)/2,sqrt(2)/2+(i)*(-(sqrt(2)/2)), -sqrt(2)/2+(i)*sqrt(2)/2,-sqrt(2)/2+(i)*(-(sqrt(2)/2))]
   The roots depends on √2. Uncheck complex mode in the Cas configuration and input :
   factor(x`^`4+1,sqrt(2))
   Output :
   (x`^`2+sqrt(2)*x+1)*(x`^`2+(-(sqrt(2)))*x+1)
   To factorize over ℂ, check complex in the cas configuration or input cFactor(x`^`4+1,sqrt(2)) (cf cFactor).