Definition of Finite Field: Addition and Multiplication Groups

[Greg Bernhardt]

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



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


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[fresh_42]

E. H. Moore probably coined in 1893 the English term Galois field in honor of Évariste Galois, who has already calculated with certain imaginary numbers modulo $p$.

The Wedderburn Theorem states that multiplication in a finite division ring is necessarily commutative. That is, finite division rings are always finite fields.

Gauss had already shown that one can count on numbers modulo a prime "as with rational numbers". Galois introduced imaginary numbers into calculation modulo $p$, much like the imaginary unit $\mathrm {i}$ in the complex numbers. He was probably the first to consider field extensions of $\mathbb{F}_ {p}$ - although the abstract concept of fields was first introduced by Heinrich Weber in 1895 and Frobenius was the first to introduce it 1896 as extensions to finite structures.