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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 11T: Finite fields and commutative rings: number-theoretic aspects

Introduction

History

Applications and related fields

For polynomials over finite fields see the page on polynomials

Subfields

• 11T06: Polynomials
• 11T22: Cyclotomy
• 11T23: Exponential sums
• 11T24: Other character sums and Gauss sums
• 11T30: Structure theory
• 11T55: Arithmetic theory of polynomial rings over finite fields
• 11T60: Finite upper half-planes [new in 2000]
• 11T71: Algebraic coding theory; cryptography
• 11T99: None of the above but in this section

Parent field: 11: Number Theory

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

McEliece, Robert J.: "Finite fields for computer scientists and engineers", Kluwer Academic Publisher, Boston, Mass., 1987. 207 pp. ISBN 0-89838-191-6

Lang, Serge: "Cyclotomic fields I and II", With an appendix by Karl Rubin. Graduate Texts in Mathematics, 121. Springer-Verlag, New York-Berlin, 1990. 433 pp. ISBN 0-387-96671-4

There is a journal, Finite Fields and Their Application.

See also the references for number theory in general.

Software and tables

Other web sites with this focus

Selected topics at this site

• Elementary remarks about solving equations mod N.
• Sample system of polynomial equations to solve modulo n
• Solving a^2+b^2+1=a mod 2^r.
• Fastest modular multiplications (summary, lit review)
• Wolstenholme's congruence (sum 1/k = 0 mod p^2)
• Citation: finding primitive roots in finite fields.
• Is there an easy way to calculate primitive roots?
• Artin's conjecture: any positive nonsquare is a primitive root for infinitely many primes. (Open)
• Is a a primitive root for infinitely many primes?
• Heath-Brown's theorem on primitive roots.
• Computing modular square roots.
• Tonelli's method of extracting square roots mod p
• Computing modular square roots
• Solving x^2+y^2+1=0 mod p efficiently
• How many solutions to x^3=2 mod p? (Class field theory) [Noam Elkies]
• What is the complexity of the discrete log calculation?
• Peculiar set of equations equivalent to : solve x^2=-x mod n
• How many quadratic residues in a row mod p? (Lit review)
• The Law of Quadratic Reciprocity
• Proof of the quadratic reciprocity theorem.
• Proving quadratic reciprocity with Stokes' Theorem
• Citations: when are specific small numbers perfect powers mod p?
• Quartic reciprocity
• Comparisons between Jacobi, Kronecker, and Legendre symbols.
• Software announcement: ZEN, for finite fields and related topics.