Linked from

The 51 pages that link to Finite field, each with the reason it gives.

• Modular arithmeticBroader topic: Residues modulo a prime provide the simplest finite fields.

• Error-correcting codeNarrower topic: Many practical codes perform their arithmetic over finite fields.

• Incidence geometryRelated: Incidence questions over finite fields have algebraic constraints and bounds distinct from real geometry.

• Field (mathematics)Broader topic: Finite fields show that field arithmetic can work on a finite set.

• Field extensionBroader topic: Finite extensions of prime fields produce every finite field.

• André WeilRelated: Counting points over finite fields is central to the results bearing Weil’s name.

• Finite setRelated: Its underlying finite set supports algebraic operations used in coding and cryptography.

• Root of unityRelated: Roots of unity in finite fields support transforms and cyclic multiplicative subgroups.

• Cyclotomic polynomialRelated: Reducing cyclotomic polynomials modulo primes helps study roots of unity in finite fields.

• Elliptic-curve cryptographyNarrower topic: Curve coordinates and arithmetic are computed in a finite field.

• Galois theoryRelated: Its extensions have explicitly describable cyclic Galois groups generated by Frobenius.

• Quotient ringRelated: Quotienting a polynomial ring by an irreducible polynomial can produce one.

• Splitting fieldRelated: Splitting fields of polynomials over finite fields are finite fields.

• Algebraic closureCompared with: No finite field is algebraically closed, though its closure is the union of finite extensions.

• FieldBroader topic: Finite fields show that fields need not contain infinitely many elements.

• Legendre symbolRelated: Residue classes modulo p form the finite field in which quadratic solvability is tested.

• Reed–Solomon codeNarrower topic: Reed–Solomon symbols and polynomial coefficients are elements of a finite field.

• Schur's theoremRelated: Schur used finite-field arithmetic to connect hypothetical Fermat solutions with a monochromatic sum.

• Coding theoryNarrower topic: Many algebraic codes perform their calculations over finite fields.

• Division ringBroader topic: Every finite division ring is a finite field, despite division rings generally allowing noncommutativity.

• Frobenius endomorphismBroader topic: On a finite field, injectivity forces Frobenius to be an automorphism.

• Hilbert's NullstellensatzCompared with: Finite fields are not algebraically closed, so maximal ideals need not arise from points over that field.

• Euler's criterionRelated: The theorem's arithmetic takes place in the field of residues modulo p.

• Discrete logarithmNarrower topic: Many classic discrete logarithm systems use the multiplicative group of a finite field.

• Boolean ringCompared with: Among finite fields, only the two-element field has every element idempotent.

• Field automorphismBroader topic: Its automorphisms are generated by the Frobenius map, giving an explicit model of the theory.

• Finite cyclic groupRelated: The nonzero elements of every finite field form a cyclic group under multiplication.

• Finite geometryRelated: Vector spaces over finite fields provide standard coordinates for finite geometries.

• Gaussian binomial coefficientNarrower topic: The coefficient counts subspaces over these fields, whose size supplies its parameter.

• Weil conjecturesNarrower topic: The conjectures concern varieties defined over fields of this kind.

• AESNarrower topic: AES performs key parts of its byte arithmetic in a finite field.

• Arithmetic geometryRelated: Counting solutions over finite fields often reveals information about varieties over number fields.

• Primitive element theoremCompared with: Finite fields are perfect, so every finite extension still has a primitive element despite lacking infinitely many choices.

• Cauchy–Davenport theoremRelated: The residues modulo a prime form the field in which the theorem operates.

• Gilbert–Varshamov boundNarrower topic: The q-ary form counts strings over an alphabet whose size is a prime power.

• Singleton boundRelated: Linear codes commonly use finite fields as their alphabets.

• Wedderburn's little theoremRelated: The established theory of finite fields provides the theorem's destination and context.

• Elwyn BerlekampRelated: Finite-field arithmetic underlies the algebraic codes Berlekamp helped design and decode.

• Freshman's dreamBroader topic: Finite fields of characteristic p exhibit the identity through their Frobenius maps.

• Hamming boundRelated: The alphabet size sets the number of words in the ambient space being packed.

• Mutually orthogonal Latin squaresRelated: Arithmetic in finite fields constructs large families of mutually orthogonal squares.

• Reed–Solomon error correctionNarrower topic: Reed–Solomon symbols are field elements, so their arithmetic wraps around a finite set.

• Adi ShamirRelated: Secret-sharing arithmetic is performed over a finite field to keep shares bounded and exact.

• Agrawal's conjectureRelated: For prime n, residues modulo n form a field, underlying the identity's behavior.

• Artin's conjecture on primitive rootsNarrower topic: Nonzero elements of a prime field form the group in which primitive roots operate.

• Ax–Grothendieck theoremRelated: Finite-field reductions turn injectivity into surjectivity for point sets.

• Bruck–Ryser–Chowla theoremCompared with: Finite fields construct projective planes of prime-power order, while this theorem mainly excludes candidate orders.

• Carmichael's theoremBroader topic: Its nonzero elements form a cyclic group by the theorem.

• Chevalley–Warning theoremNarrower topic: The theorem’s divisibility conclusion uses the characteristic of this field.

• Hasse's theorem on elliptic curvesNarrower topic: The theorem's parameter q is the number of elements in this field.