3. Variables on Galois fields

The Galois Field Package has two kinds of variables over finite fields. One is a variable which represents an element of a field. When we see in a mathematics book "for an element x of the field K" , the symbol x represents a constant of K. This variable works like an element of K and may be substituted by a constant later. The second kind of variable is called an indeterminate or an algebraic variable. The algebraic variable can never be substituted by a constant and dose not work like an element.

The following is a well known theorem on the Galois field:

Theorem 1. For any element c of GF(q), c^q=c.

This theorem works for a variable which represents an element but not for an algebraic variable. When we declare variables which represent elements of the finite field K9.

DeclareVariables[K9,x,y]

Since K9 is a finite field of order 9, the theorem 1 works like

x^9

x

x^-10

  -2
x

From the theorem 1, x⁹ = x in GF(9). Exponents of variable x in outputs can never be larger than 8 ( generally "order" - 1).

you can see algebra name in which a variable was declared by the function AlgebraName.

AlgebraName[x^2-1]

K9

Mathematica does not print in the form " x⁰ ". Therefore, the Galois Field Package prints with the form
x ^Zero. The symbol Zero exactly works as the numeric number 0.

x^0

 Zero
x

% /. x-> K9[5]

K9[1]

Theorem 2. For any nonzero element d of GF(q), d^(q-1) = 1.

The theorem 2 says that if x is an nonzero element of K9 then x⁸=1. The Galois Field Package have a declaration function for nonzero variable.

NonZero[x]

x^{0,1,2,3,4,5,6,7,8,9,10}

            2   3   4   5   6   7             2
{K9[1], x, x , x , x , x , x , x , K9[1], x, x }

You can compare with a ordinal variable y of K9.

y^{0,1,2,3,4,5,6,7,8,9,10}

  Zero      2   3   4   5   6   7   8      2
{y    , y, y , y , y , y , y , y , y , y, y }

{x , y }^16

         8
{K9[1], y }

Note that 0-th power of 0 is indeterminate.

K9[0]^0

Power:K9[0]^-1 and K9[0]^0 are undefined

Indeterminate

Nonzero variables can also be declared by the NonZero option in the function DeclareVariables.

DeclareVariables[K9,x, NonZero -> True]

Inverse[{{ x,1},{0,x}}]

  1  K9[2]       1
{{-, -----}, {0, -}}
  x    2         x
      x

Next, when we declare an algebraic variable, we use the function DeclareAlgebraicVariables or DeclareIndeterminates.

DeclareAlgebraicVariables[K9,X,Y]

Using the algebraic variable, we can define a polynomial ring GF(9)[X]. Since algebraic variables are not elements of a finite field, the theorem 1 and 2 have no effect on X and Y.

X^9

 9
X

We define the 0-th power of an algebraic variable to be 1.

X^0

K9[1]

Since X, Y are indeterminate, you can not substitute a constant to these variables.

X = K9[2]

Set::wrsym: Symbol X is Protected.

K9[2]

The Factor function of a polynomail over a Galois field is contained in the subpackage Factor.m

<<Factor.m

Factor.m was loaded.

Factor[X^7-1]

              3            2
(X + K9[2]) (X  + K9[2] + X  K9[3] + X K9[5]) 
 
    3                      2
  (X  + K9[2] + X K9[6] + X  K9[7])