##
**Infinite algebraic extensions of finite fields.**
*(English)*
Zbl 0674.12009

Contemporary Mathematics, 95. Providence, RI: American Mathematical Society (AMS). xv, 104 p. $ 24.00 (1989).

The declared aim of the authors of this monograph is to show that one can view much of the theory of finite algebraic extensions of finite fields in such a way that the theory can be extended to infinite algebraic extensions. The work is aimed at graduate students and advanced undergraduates with some background of abstract algebra. This gathering together of results in the area of infinite algebraic extensions is timely because of the current interest in applications of finite field theory in graph theory, cryptography, coding theory, statistical designs and other areas of discrete mathematics.

The monograph begins with a résumé of the expected classical results on the nesting of finite fields \(\mathrm{GF}(q)\) with the same characteristic \(p\), together with polynomial representations of functions \(\mathrm{GF}(q)\to \mathrm{GF}(q)\), \(p\)-polynomials over \(\mathrm{GF}(q^n)\), Dickson polynomials with criteria for them to be permutation polynomials. On passing to infinite algebraic extensions of \(\mathrm{GF}(q)\) emphasis is put on the usefulness of the lattice of Steinitz numbers \(N=\prod p_i^{x_i}\) \((0\leq x_i\leq \infty\); \(p_1,p_2,\ldots\) prime numbers) in describing the nesting of such extensions \(\mathrm{GF}(q^N)\). Divisor sequences \(d_0,d_1,\ldots\), \((d_i\mid d_{i+1})\) and series \(\sum a_id_i \) \((0\leq a_i<d_{i+1}/d_i)\) are introduced as devices for the handling of the arithmetic in \(\mathrm{GF}(q^N)\) and generalized \(q\)-polynomials.

Chapter 3 attacks the problem of calculations in \(\mathrm{GF}(q^N)\) by showing how an iterated presentation \((d_i,p_i(x))\) leads to an explicit basis of \(\mathrm{GF}(q^N)\) over \(\mathrm{GF}(q)\) and proceeds to give that iterated presentation due to D. Wiedemann [Fibonacci Q. 26, No. 4, 290–295 (1988; Zbl 0658.12012)] when \(q=2\) and \(N=2^{\infty}\), generalizes that of J. H. Conway [On numbers and games. New York etc.: Academic Press (1976; Zbl 0334.00004)] for the same \(q, N\) to the case when \(q=p^N\) and \(N=p^{\infty}\) (\(p\) any prime) and deals with a class of cases with \(N=(p-1)p^{\infty}\) (\(p\) not the radical of \(q\)).

Chapter 4 begins with polynomials and polynomial functions of \(\mathrm{GF}(q^N)\) and gives necessary and sufficient conditions for monomials, \(q\)-polynomial and Dickson polynomials to be permutation polynomials so extending the classical results of the first chapter. With \(N\) infinite there are, of course, functions from \(\mathrm{GF}(q^N)\) to itself which have no polynomial representation. However the notion of polynomial can be extended via a divisor sequence to ensure that all functions are representable by such extended polynomials. In particular all linear transformations of \(\mathrm{GF}(q^N)\) over \(\mathrm{GF}(q)\) can be represented by extended \(q\)-polynomials.

In the final chapter two applications are given. The first shows that \(\mathrm{GF}(q^N)\) can be used to construct complete sets of infinite Latin squares. The second one deals with the construction of nonlinear polynomials which by substitution give permutations of the set of \(m\times m\) matrices over \(\mathrm{GF}(q^N)\).

The reader will find the inclusion of several examples helpful particularly in illustrating the later results, just as he will find the lack of an index unhelpful.

The monograph begins with a résumé of the expected classical results on the nesting of finite fields \(\mathrm{GF}(q)\) with the same characteristic \(p\), together with polynomial representations of functions \(\mathrm{GF}(q)\to \mathrm{GF}(q)\), \(p\)-polynomials over \(\mathrm{GF}(q^n)\), Dickson polynomials with criteria for them to be permutation polynomials. On passing to infinite algebraic extensions of \(\mathrm{GF}(q)\) emphasis is put on the usefulness of the lattice of Steinitz numbers \(N=\prod p_i^{x_i}\) \((0\leq x_i\leq \infty\); \(p_1,p_2,\ldots\) prime numbers) in describing the nesting of such extensions \(\mathrm{GF}(q^N)\). Divisor sequences \(d_0,d_1,\ldots\), \((d_i\mid d_{i+1})\) and series \(\sum a_id_i \) \((0\leq a_i<d_{i+1}/d_i)\) are introduced as devices for the handling of the arithmetic in \(\mathrm{GF}(q^N)\) and generalized \(q\)-polynomials.

Chapter 3 attacks the problem of calculations in \(\mathrm{GF}(q^N)\) by showing how an iterated presentation \((d_i,p_i(x))\) leads to an explicit basis of \(\mathrm{GF}(q^N)\) over \(\mathrm{GF}(q)\) and proceeds to give that iterated presentation due to D. Wiedemann [Fibonacci Q. 26, No. 4, 290–295 (1988; Zbl 0658.12012)] when \(q=2\) and \(N=2^{\infty}\), generalizes that of J. H. Conway [On numbers and games. New York etc.: Academic Press (1976; Zbl 0334.00004)] for the same \(q, N\) to the case when \(q=p^N\) and \(N=p^{\infty}\) (\(p\) any prime) and deals with a class of cases with \(N=(p-1)p^{\infty}\) (\(p\) not the radical of \(q\)).

Chapter 4 begins with polynomials and polynomial functions of \(\mathrm{GF}(q^N)\) and gives necessary and sufficient conditions for monomials, \(q\)-polynomial and Dickson polynomials to be permutation polynomials so extending the classical results of the first chapter. With \(N\) infinite there are, of course, functions from \(\mathrm{GF}(q^N)\) to itself which have no polynomial representation. However the notion of polynomial can be extended via a divisor sequence to ensure that all functions are representable by such extended polynomials. In particular all linear transformations of \(\mathrm{GF}(q^N)\) over \(\mathrm{GF}(q)\) can be represented by extended \(q\)-polynomials.

In the final chapter two applications are given. The first shows that \(\mathrm{GF}(q^N)\) can be used to construct complete sets of infinite Latin squares. The second one deals with the construction of nonlinear polynomials which by substitution give permutations of the set of \(m\times m\) matrices over \(\mathrm{GF}(q^N)\).

The reader will find the inclusion of several examples helpful particularly in illustrating the later results, just as he will find the lack of an index unhelpful.

Reviewer: David Kirby (Southampton)

### MSC:

12E20 | Finite fields (field-theoretic aspects) |

11T06 | Polynomials over finite fields |

12E05 | Polynomials in general fields (irreducibility, etc.) |

12F05 | Algebraic field extensions |

12-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory |

11T71 | Algebraic coding theory; cryptography (number-theoretic aspects) |