## Fields of Beer February 21, 2006

A field is a *set* of elements that is *closed* under, and satisfies the axioms of, *associativity*, *commutativity*, *distributivity*, identity and *inverse*, for addition and multiplication. The number of elements in a field *Z _{p}* is

*p*, and

*p*is always prime (or a power of a prime).

A field with a finite number of elements is called a *Galois* Field. As an aside to this aside, in looking at error correcting codes, a binary code (alphabet {*0*, *1*}) with codewords of length *n* has all its codes in the field *GF*(2^{n}). *GF*(2^{n}) is a vector space of *n*-tuples over *GF*(*2*).

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