title={FINITE FIELDS},
  author={Keith Conrad},
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
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
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
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
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
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


On the product of two primitive elements of maximal subfields of a finite field
Abstract Let F r denote a finite field with r elements. Let q be a power of a prime, and p 1 , p 2 , p 3 be distinct primes. Put y 1 =p 1 p 2 , y 2 =p 1 p 3 , y 3 =p 2 p 3 , z=p 1 p 2 p 3 ,A={(t 1 ,tExpand
On decomposition of sub-linearised-polynomials
Abstract We give a detailed exposition of the theory of decompositions of linearised polynomials, using a well-known connection with skew-polynomial rings with zero derivative. It is known that thereExpand
Elements of Prescribed Order, Prescribed Traces and Systems of Rational Functions Over Finite Fields
  • F. Özbudak
  • Computer Science, Mathematics
  • Des. Codes Cryptogr.
  • 2005
It is proved that for all sufficiently large extensions of F, there is an element $\xi \in {\mathbb F}_{q^{km}}$ of prescribed order such that (\xi)=\gamma_i$ for $i=1, \ldots, r$. Expand
A computational introduction to number theory and algebra
  • V. Shoup
  • Computer Science, Mathematics
  • 2005
This edition now includes over 150 new exercises, ranging from the routine to the challenging, that flesh out the material presented in the body of the text, and which further develop the theory and present new applications. Expand
On the sum of two primitive elements of maximal subfields of a finite field
Let F"r denote a finite field with r elements. Let q be a power of a prime, and p"1,p"2, p"3 be distinct primes.Expand
Mathematical Papers read at the International Mathematical Congress held in connection with the World's Columbian Exposition, Chicago, 1893
THIS book, which is an excellent specimen of mathematical printing, constitutes vol. i. of “Papers published by the American Mathematical Society.” The 400 pages contain thirty-nine papers. GermanExpand
Abstract Algebra
Abstract AlgebraBy Andrew O. Lindstrum jun. (Holden-Day Series in Mathematics.) Pp. xii + 211. (San Francisco and London: Holden-Day, Inc., 1967.) $10.
Cours d'algèbre
Cours d'algèbre, Nouvelle Bibliothèque Mathématique [New Mathematics Library
  • Cours d'algèbre, Nouvelle Bibliothèque Mathématique [New Mathematics Library
  • 1997
Niederreiter – Introduction to finite fields and their applications
  • 1994