1. A Galois field divider engine for providing division operations for forward error correction and/or error detection over a Galois field, the Galois field divider engine comprising: a Galois field reciprocal generator circuit, said Galois field reciprocal generator circuit including first and second Galois field multiplier circuits, an output of said first Galois field multiplier circuit fed to both inputs of said second Galois field multiplier circuit to provide the square of the output of said first Galois field multiplier circuit; an input selection circuit for initially inputting a 1 and a first Galois field element to the Galois field reciprocal generator circuit to obtain an output, subsequently multiplying in the Galois field reciprocal generator circuit the first Galois field element by the output of the second Galois field reciprocal generator circuit for predicting a modulo remainder of the multiply-square of the polynomial product of an irreducible polynomial m−2 times where m is the degree of the Galois field, to obtain the reciprocal of the first Galois field element and multiplying in the Galois field divider engine the reciprocal of the first Galois field element by a second Galois field element for predicting a modulo remainder of the polynomial product for an irreducible polynomial to obtain the quotient of the two Galois field elements in m cycles.