This book is a first step in a new direction: to modify existing theory from a constructive point of view and to stimulate the readers to make their own computational experiments. We are thoroughly convinced that their observations will help to build a new basis from which to venture into new theory on algebraic numbers. History shows that in the long run, number theory always followed the cyclic movement from theory to construction to experiment to conjecture to theory.

Consequently, this book is addressed to all lovers of number theory. On the one hand, it gives a comprehensive introduction to (constructive) algebraic number theory and is therefore especially suited as a textbook for a course on that subject. On the other hand, many parts go far beyond an introduction and make the user familiar with recent research in the field. For experimental number theoreticians we developed new methods and obtained new results (e.g., in the tables at the end of the book) of great importance for them. Both computer scientists interested in higher arithmetic and in the basic makeup of digital computers, and amateurs and teachers liking algebraic number theory will find the book of value.

Many parts of the book have been tested in courses independently by both authors. However, the outcome is not presented in the form of lectures, but, rather, in the form of developed methods and problems to be solved. Algorithms occur frequently throughout the presentation.

Introduction

Since the first printing of this book in 1989 algorithmic algebraic number theory has attracted rapidly increasing interest. This is documented, for example, by a regular meeting, ANTS (algebraic number theory symposium), every two years whose proceedings give a good survey about ongoing research. Also there are several computer algebra packages concentrating on number theoretical computations. At present the most prominent ones, which are available for free, are KANT, PARI and SIMATH. KANT comes with a data base for algebraic number fields, already containing more than a million fields of small degree. KANT is developed by the research group of the author at Berlin and will be presented in some detail in this chapter. We note that almost all of KANT and PARI is also contained in the MAGMA system.

In the sequel we shortly discuss the improvements which were obtained for the computation of the important invariants of algebraic number fields. On the other hand, in computational algebraic number theory the interest has gradually turned from absolute extensions to relative extensions of number fields and we will sketch the important developments in that area. If subfields exist, the information about the invariants of those subfields can be lifted and used in the field under consideration. This relative point of view permits computations in fields of much larger degrees and has important applications, for example to class field computations.

Transistors were invented at just the right time for information technology. Offering high reliability and low power consumption, they were immediately attractive to telecommunication engineers and computer designers, and their use surged ahead when it became possible to manufacture thousands of transistor circuits on one small wafer of silicon. Intense competition between suppliers forced them into continuous improvement and, because costs fell sharply with mass production, competition led also to the creation of surplus manufacturing capacity. Producers scrambled to find new markets to absorb their rising output, and few corners of life in Western countries have been left untouched by the silicon chip.

Technological developments of every kind have been so rapid in this century that we have had to accept more changes within our lifetimes than once were spread over many generations. Information technology has helped to force the pace of change in industry, commerce, government and everyday living; but nowhere is development faster than in information technology (IT) itself. Genetic engineering, nuclear energy, and IT have been lumped together as extreme examples of our obsession with ‘high technology’. Critics have been moved to speculate whether changes are now coming upon us too rapidly to be accommodated without unacceptable amounts of human and social stress.

Many forecasts of life under IT strive to make our flesh creep at the prospect of a bleak future among the robots in a police state.

It is convenient to consider the ways in which IT may be used to threaten or to protect the safety and security of our lives and property under four heads: safety of life, invasion of privacy, crime and war. Malign intent is not the principal reason for concern in the Western democracies, but risks can arise when IT systems themselves are faulty. In the early days, failures were mainly caused by hardware breakdowns; today, silicon chips are highly reliable, and the emphasis has moved to data errors and software ‘bugs’.

Input data may be incorrectly scanned by automatic readers, and human operators make mistakes when entering data through a keyboard; checks are therefore made to reveal the presence of errors. The simplest, but not the cheapest, check is for two independent operators to enter the data and then to compare their versions; if they agree then it is highly likely that no mistake was made. Otherwise, the entry is repeated until they do agree. Or, a batch of items of numerical data may be added together on a pocket calculator and the total entered as a ‘sum check’ to be compared with the result of the same addition by the computer once input is complete. It is not necessary for the total to have a meaning. Thus, the birth dates of a group of staff can be totalled for comparison; such a total is known as a ‘hash total’.

The age of information

It would be boring indeed to detail the innumerable ways in which information has become important to economic activity and social cohesion. We have all been told so many times. If information really does perform so vital a function it must be very different from the disposable stuff which pours over us in an unending mish-mash of news, views and abuse. Facts, speculations and persuasion are smoothly blended – I almost wrote ‘blanded’ – as the trite, the trivial and the titillating are fleetingly presented as having as much claim on our attention as more important matters. We have no control over the flow, and no way of answering back. We are, of course, still allowed to turn it off, but like amputation that is a remedy of the last resort.

Perhaps the development of IT will let us select, question and compare. It is necessary to write ‘perhaps’ because IT merely enables, it does not compel and cannot guarantee. Strong commercial and political interests will continue to fish for our attention and strive to steer our responses. This they would be able to do all the more insidiously were we to permit our use of IT to persuade us that we were now in full control. What can come out of an IT system depends on what goes into it, and I see no rush to abandon control over the primary sources.

Programmed control

Marshal McLuhan could be lured by the prospect of an epigram into uttering a delphic half-truth. Even so, his phrase, ‘the medium is the message’, does actually fit a large IT system, for its imposing façade confers an undeserved authority on its output. We charitably assume that so much blood, brains and treasure would not have been poured out unless the results commanded our instant and unquestioning acceptance. But it is not quite like that. Computing plus communications is a quite exceptionally powerful combination, and we need to consider why that should be so. For most people, computers are the mysterious part of IT. After more than a century's experience, rapid electrical communications over long distances are taken very much for granted.

With the widespread use of personal computers, everyone now knows that ‘hardware’ is the electronic and mechanical equipment, and ‘software’ the controlling programs which make it do what we require. Interchangeable control is not a particularly new idea. The drive mechanisms, electronics and loudspeakers of a record player are the hardware of a general-purpose musical instrument which can simulate a symphony orchestra or a soprano by playing the corresponding record. Records are easily changed, and there is no limit to the number of different ones that can be played. It is the same with computers and their programs.

1. Universities and research laboratories

1.1. Conduct research free from external misdirection in pursuit of short-range objectives set by industry or the state (8.3, below).

1.2. Develop an improved theoretical base for cooperative multiprocessor computing in parallel and network systems.

1.3. Inform the professions and the media fully and frequently about research in progress and its implications.

2. The IT industry

2.1. Continue free to develop and produce hardware, software and services to meet commercial objectives of own choosing, subject only to the trading controls generally applied to protect the rights of customers.

2.2. Participate in,v and help to fund, the development of improved methods of technology assessment (4.4 and 8.5, below).

3. Users of IT systems

3.1. Continue free to develop and operate IT systems at will, scrupulously observing legislation to protect privacy, and subject only to the controls generally applied to maintain the rights of customers and employees.

3.2. Top management and governing boards: learn enough about IT to exercise firm strategic control over their organization's use of it, in order to redirect projects (too narrowly conceived by specialists) which could damage their customer or industrial relations (4.5, below).

3.3. Take potential consequences into account when designing systems, accepting some extra cost, delay and loss of efficiency where necessary to mitigate adverse effects (4.5, below).

3.4. Consult employees and customers liable to be affected by new proposals before these have been finalized, and while they can be modified to meet valid objections.