Finite field Finite ring

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the theory of finite fields perhaps important aspect of finite ring theory due intimate connections algebraic geometry, galois theory , number theory. important, old aspect of theory classification of finite fields (jacobson 1985, p. 287):



the order or number of elements of finite field equals p, p prime number called characteristic of field, , n positive integer.
for every prime number p , positive integer n, there exists finite field p elements.
any 2 finite fields same order isomorphic.

despite classification, finite fields still active area of research, including recent results on kakeya conjecture , open problems regarding size of smallest primitive roots (in number theory).


a finite field f may used build vector space of n-dimensions on f. matrix ring of n × n matrices elements f used in galois geometry, projective linear group serving multiplicative group of a.







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