= Finite field
{title2=$\mathbb F_q$}
{wiki}
= Galois field
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{synonym}
A <finite field> is a <field> with finitely many elements: <addition>, <subtraction>, <multiplication>, and <division> by a nonzero element all stay inside a finite collection. Write $\mathbb F_q$, or $\operatorname{GF}(q)$, for a <finite field> with $q$ elements. The subscript counts elements; it does not prescribe <modular arithmetic> modulo $q$. The <field with four elements> is the smallest example where that distinction matters.
Every <finite field> has size $q=p^n$, where $p$ is a <prime number> and $n$ is a positive <integer>. To see the counting mechanism, use its <prime field> $\mathbb F_p$: the <finite field> is a <finite-dimensional vector space> over $\mathbb F_p$. For a <basis> $e_1,\ldots,e_n$, every element has exactly one expression
$$
c_1e_1+\cdots+c_ne_n,\qquad c_i\in\mathbb F_p.
$$
Each <coefficient> has $p$ choices, so there are $p^n$ expressions. Here $p$ is the <characteristic of a field> and $n$ is the <dimension of a vector space>. Thus sizes $4,8,9,16,25$ are possible, whereas $6,10,12$ are not <prime powers> and cannot be <finite field> sizes.
Conversely, every <prime power> occurs, and any two <finite fields> of the same size admit a <field isomorphism>. These are classification statements, beyond the counting argument. See Conrad's Theorems 1.5, 2.2, and 2.7.https://kconrad.math.uconn.edu/blurbs/galoistheory/finitefields.pdf[Keith Conrad, Finite Fields, sections 1–2]{ref} Uniqueness means the <arithmetic> agrees after relabeling; it does not choose a <basis>, an <irreducible polynomial> for a <quotient ring> representation, or a particular <field isomorphism>. Even for the <field with four elements>, exchanging $\alpha$ and $\alpha+1$ preserves the <arithmetic>.
The appeal for <coding theory> is exact, finite <arithmetic> with reliable <division>: <polynomial interpolation> and <linear algebra> work over a finite alphabet. A <Reed-Solomon code> represents messages by <polynomials> and adds redundancy by evaluating them at several distinct <finite field> elements. Forney's MIT notes develop this <arithmetic> as preparation for such <error-correcting codes>.https://ocw.mit.edu/courses/6-451-principles-of-digital-communication-ii-spring-2005/8921c013c1d7802bb197572c52442b97_chap7.pdf[G. David Forney Jr., Introduction to finite fields, 2005]{ref} In <cryptography>, <finite fields> likewise supply exact <arithmetic> on finitely many symbols, with operations that can be implemented using <bits>.
Historically, Évariste Galois published work on <finite fields> beyond <prime fields> in 1830, constructing them by adjoining elements satisfying <irreducible polynomials> over <prime fields>. Conrad's historical account also records E. H. Moore's 1893 terminology, from which the name <Galois field> and notation $\operatorname{GF}(q)$ descend.https://kconrad.math.uconn.edu/math5211s13/handouts/finitefield.pdf[Keith Conrad, Finite Fields, section 5: History]{ref}
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