6.34.8  Division in ℤ/pℤ or in ℤ/pℤ[x]: /

The / operator divides two integers in ℤ/pℤ or two polynomials A and B in ℤ/pℤ[x]. (See also Section 6.8.2.) Since ℤ/pℤ is only a field if p is prime, the quotient is only guaranteed to exist if p is prime (unless the denominator is 0 (mod p )).

• For integers in ℤ/pℤ:

  
  

  Example.

  • Input:
    5%13/2% 13
    Since 13 is prime, you get:
    Output:

    ⎛ ⎝

    −4

    ⎞ ⎠

    %13

  • Input:
    5%14/3% 14
    Since 3(mod 1 )4 is invertible in Z/14ℤ, you get:
    Output:

    ⎛ ⎝

    −3

    ⎞ ⎠

    %14

  • Input:
    5%14/7% 14
    Since 7(mod 1 )4 is not invertible in Z/14ℤ, you will get an error: Output:
    Not invertible Error: Bad Argument Value
• For polynomials, the result of P/Q is its irreducible representation in ℤ/pℤ[x].

  
  

  Example.
  Input:

  (2*x^2+5)%13/(5*x^2+2*x-3)%13

  Output:

  ⎛ ⎝ 6%13 ⎞ ⎠ x+1%13 ⎛ ⎝ 2%13 ⎞ ⎠ x+ ⎛ ⎝ 2%13 ⎞ ⎠ %13