# FINITE FIELDS

@inproceedings{Conrad2004FINITEF,
title={FINITE FIELDS},
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
2,160 Citations
A note on non-recurring sequences over Galois fields
Let F be a Galois field and Γ(F ) be the F [D]–module of all sequences over F , [4]. Consider an f(D) 6= 0 in F [D]. The concept of a pseudoperiodic sequence with f(D) as its pseudo-characteristicExpand
On the graph of a function over a prime field whose small powers have bounded degree
• Computer Science, Mathematics
• Eur. J. Comb.
• 2009
The conjecture that the graph of f is contained in an algebraic curve of degree t-1 is conjecture and the conjecture for t=2 and t=3 is proved and the results apply to functions that determine less than p-2p-1+114 directions. Expand
A decade of Finite Fields and Their Applications
• G. Mullen
• Computer Science, Mathematics
• Finite Fields Their Appl.
• 2005
The aim of this article is to formulate the bounds on plane (n, r)-arcs as bounds that look familiar to coding theorists, to survey recent improvements, and to list a number of open problems. Expand
A short proof for explicit formulas for discrete logarithms in finite fields
• H. Niederreiter
• Mathematics, Computer Science
• Applicable Algebra in Engineering, Communication and Computing
• 2005
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 groupExpand
Nonstandard linear recurring sequence subgroups in finite fields and automorphisms of cyclic codes
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
A note on the irreducibility of polynomials over finite fields
It is difficult in general to determine whether a given polynomial is irreducible. However, for polynomials over a finite field, various irreducibility criteria were proposed (details of which can beExpand
Functions over finite fields that determine few directions
• Simeon Ball
• Computer Science, Mathematics
• Electron. Notes Discret. Math.
• 2007
Abstract We investigate functions f over a finite field F q , with q prime, with the property that the map x goes to f ( x ) + c x is a permutation for at least 2 q − 1 elements c of the field. WeExpand
Irreducibility and Deterministic r-th Root Finding over Finite Fields
• Mathematics, Computer Science
• Electron. Colloquium Comput. Complex.
• 2017
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
Spectrally arbitrary patterns over finite fields
• Mathematics
• 2012
An n × n zero–nonzero pattern 𝒜 is spectrally arbitrary over a field 𝔽 provided that for each monic polynomial r(x)∈𝔽[x] of degree n, there exists a matrix A over 𝔽 with zero–nonzero pattern 𝒜Expand
On the existence of some specific elements in finite fields of characteristic 2
• Computer Science, Mathematics
• Finite Fields Their Appl.
• 2012
This paper considers the existence of some specific elements in F q n, the finite field with q n elements, and finds an element ξ in Fq n such that ξ is a primitive normal element and ξ + ξ − 1 is a primordial element of FQ n. Expand

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