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When M is finite, say M = {α1 , . . , αk }, we write K(α1 , . . , αk ) for K(M ). 57 Definition Let K ⊆ F , α ∈ F , and f (α) = 0 where f is a monic polynomial in K[x]. Then f is the minimal polynomial of α if α is not a root of any nonzero polynomial in K[x] of lower degree. 58 Proposition The minimal polynomial of any extension field element is irreducible over the base field. This result provides a method by which one can obtain irreducible polynomials. 59 Definition A field F is a finite extension of K if K ⊆ F and F is a finite dimensional vector space over K.

Then every nonzero element of Fq can be written as a power of θ. This representation makes multiplication of field elements Introduction to finite fields 17 very easy to compute. 5. Conversely, as we will see later in our discussion of bases for finite fields, representations which make exponentiation easy to compute often have a more complex multiplicative structure. 40 Definition Let α ∈ F∗q . The order of α is the smallest positive integer n such that αn = 1. 41 Remark We use the notation (a, b) or gcd(a, b) to represent the greatest common divisor (gcd) of a and b, where a and b belong to a Euclidean domain (usually integers or polynomials).

A. P. uk Xiang-dong Hou Department of Mathematics and Statistics University of South Florida 4202 E. A. edu W. Cary Huffman Department of Mathematics and Statistics Loyola University Chicago 1032 W. A. edu Michael Jacobson, Jr. A. O. no Melsik Kyuregyan Institute for Informatics and Automation Problems National Academy of Sciences of Armenia 1, P. O. A. nl xxxiii Contributors Antonio Rojas-Le´ on Gary L. Mullen ´ Departamento de Algebra - Facultad de Department of Mathematics Matem´ aticas The Pennsylvania State University Universidad de Sevilla University Park, PA 16802 Apdo.

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