Extending the "integer" concept to algebraic numbers suggests the more general algebraic concept of ring. Likewise the concept of rational number suggests the algebraic concept of field. In this chapter we look specifically at fields of algebraic numbers and how to define their "integers." This involves the study of polynomial rings and the corresponding concepts of "prime" polynomial and "congruence modulo a prime." Then we return to algebraic number fields and view them "relative to" their subfields, such as the fields of rational numbers. This is facilitated by ideas from linear algebra, such as basis and dimension.

The chapter is mostly about combinatorics on words, an important topic since many algorithms are based on combinatorial properties of their input. Several problems are related to periodicity in words, which is a major combinatorial tool in many algorithms presented in following chapters. The stringologic proof of Fermat’s little theorem, codicity testing, distinct periodic words, and problems about conjugate words are introductory problems in applications of periodicities. Then a couple of problems related to famous abstract words: Fibonacci, Thue-Morse and Oldenburger- Kolakoski sequences are presented. They are followed by some algorithmic constructions of certain special supersequences and superwords as well of interesting classes of words: Skolem and Langford sequences. Many problems in this chapters are of algorithmic and constructive type.

There is a proliferation of methods of point estimation other than ML. First, MLEs may not have an explicit formula and may be computationally more demanding than alternatives. Second, MLEs typically require the specification of a distribution. Third, optimization of criteria other than the likelihood may have some justification. The first argument has become less relevant with the advent of fast computers, and the alternative estimators based on it usually entail a loss of optimality properties. The second can be countered to some extent with large-sample invariance arguments or with the nonparametric MLE and empirical likelihood seen earlier. However, the third reason can be more fundamental.This chapter presents a selection of four common methods of point estimation, addressing the reasons outlined earlier, to varying degrees: method of moments, least squares, nonparametric (density and regression), and Bayesian estimation methods. In addition to these reasons for alternative estimators, point estimation itself may not be the most informative way to summarize what the data indicate about the parameters. Therefore, the chapter also introduces interval estimation and its multivariate generalization, a topic that leads quite naturally to the subject matter of Chapter 14.