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What is a finite field

  1. Jul 23, 2014 #1

    All finite fields are known; they are the Galois fields GF(p^n), where p is a prime.

    They have addition group Z(p)^n and multiplication group Z(p^n-1); their multiplication groups are cyclic.

    If p = 2, then addition and multiplication can be done very fast by typical computer hardware using bitwise exclusive or and shifting.


    Extended explanation

    The finite fields GF(p) are {0, 1, ..., p-1} under addition and multiplication modulo p, which must be a prime number.

    Finite fields GF(pn) for n > 1 can be described using polynomials in a variable x with coefficients having values in {0, 1, ..., p-1}.

    Every element is a polynomial with a degree at most n-1.

    Element addition is polynomial addition modulo p, while element multiplication is polynomial multiplication modulo p and a degree-n primitive or irreducible polynomial.

    A primitive polynomial is one that cannot be factored in this construction of GF(pn). Primitive polynomials are not unique; there are
    [itex]N(p,n) = \frac{1}{n} \sum_{m|n} \mu(m) p^{n/m}[/itex]

    monic ones, where μ is the Möbius mu function. That function is (-1)number of prime factors if they all have power 1, and 0 otherwise.

    A finite field GF(pn) has subfield GF(pm) if m evenly divides n. If n is a prime, then GF(pn) only has only the trivial subfields, itself and GF(p).

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
  2. jcsd
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