Field: A set of numbers with two operations + and x, such that the addition is distributive with respect to the multiplication. Moreover, each non zero element is invertible. The set of real numbers R, the ratiional numbers Q, the complex number C are field with an infinite number of elements. When the cardiinal of the field is finite, we called it a finite field. For example, if p is a prime integers (divisible only by one and itself), then Z/pZ is a finite field. <p>

Example: p=2. F_2 = ({0,1},+ mod 2, x mod 2) is a field. Compute its addition and multiplication tables. <p>

Example: p=7, F_p=Z/pZ. Same exercice: compute its addition and multiplication tables. <p>

If n=6, Z/nZ is not a field. If one computes the multiplication table, it is easy to see that 4 is not invertible. There is no solution to the equation 4x = 3 mod 6, while such equation has always a single solution mod 7. Z/6Z is called a ring. This is also the case for Z the set of integers with + and x. The only invertible elements are -1 and 1. <p>

To compute the multiplicative inverse of an element, one usually computes the Extended Euclidean Algorithm. In the case of prime finite field, we can also use Fermat Little Theorem, which states that x^{p-1}=1 mod p for x not equal to 0, and so x^{p-2} mod p is the inverse of x mod p. <p>

Th: For every prime p, Z/pZ is a prime finite field with p elements. The set of non zero elements forms a cyclic multiplicative group.