# FINITE FIELDS

@inproceedings{Conrad2004FINITEF, title={FINITE FIELDS}, author={Keith Conrad}, year={2004} }

This handout discusses finite fields: how to construct them, properties of elements in a finite field, and relations between different finite fields. We write Z/(p) and Fp interchangeably for the field of size p. Here is an executive summary of the main results. • Every finite field has prime power order. • For every prime power, there is a finite field of that order. • For a prime p and positive integer n, there is an irreducible π(x) of degree n in Fp[x], and Fp[x]/(π(x)) is a field of order… Expand

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Let Fq be the finite field of order q and characteristic p, so that q is a power of the prime p. Then the multiplicative group F* of nonzero elements of Fq is cyclic q and a generator of this group… Expand

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The known classification of the subgroups of PGL(2,q) in combination with a recent result by Brison and Nogueira are used to show that a nonstandard element of degree two over GF(q) necessarily is of type I or type II, thus solving completely the classification problem for the case m = 2. Expand

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An extension of Stickelberger's Lemma is given; r-th nonresidues are constructed from a polynomial f for which there is a d, such that, r|d and r ł#(irreducible factors of f(x) of degree d). Expand

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