Menu Home Action My entries Defined browse Select Select in the list MathematicsPhysics Then Select Select in the list Then Select Select in the list Search

finite field

 Definition/Summary 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.

 Equations

 Scientists Évariste Galois

 Recent forum threads on finite field

 Breakdown Mathematics > Algebra >> Field Theory

 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 $N(p,n) = \frac{1}{n} \sum_{m|n} \mu(m) p^{n/m}$ 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).