Algorithms in Algebraic Number Theory

Abstract

Abstract.In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers. IntroductionThe main interest of algorithms in algebraic number theory is that they provide number theorists with a means of satisfying their professional curiosity. The praise of numerical experimentation in number theoretic research is as widely sung as purely numerological investigations are indulged in, and for both activities good algorithms are indispensable. What makes an algorithm good unfortunately defies definition-too many extra-mathematical factors affect its practical performance, such as the skill of the person responsible for its execution and the characteristics of the machine that may be used.The present paper addresses itself not to the researcher who is looking for a collection of well-tested computational methods for use on his recently acquired personal computer. Rather, the intended reader is the perhaps imaginary pure mathematician who feels that he makes the most of his talents by staying away from computing equipment. It will be argued that even from this perspective the study of algorithms, when considered as objects of research rather than as tools, offers rich rewards of a theoretical nature.The problems in pure mathematics that arise in connection with algorithms have all the virtues of good problems. They are of such a distinctly fundamental nature that one is often surprised to discover that they have not been considered earlier, which happens even in well-trodden areas of mathematics; and even in areas that are believed to be well-understood it occurs frequently that the existing theory offers no ready solutions, fundamental though the problems may be. Solutions that have been found often need tools that at first sight seem foreign to the statement of the problem.

Fast quantum algorithms for computing the unit group and class group of a number field

“…For example, Lenstra [6] points out that number fields appear in algorithms that do not even refer to number fields in the problem statement. The most notable example is probably the number field sieve, which is the best classical algorithm for factoring integers.…”

“…a finite extension of Q . For references about the computational aspects see [6,3,13]. As a field extension, F can be generated by a single element θ in C, F = Q(θ).…”

“…The ring of integers is defined as the set of numbers in F that are the root of some monic polynomial in Z[x]. Computing O is polynomial-time equivalent to finding the largest square factor of a given positive integer [6,Thm 4.4]. Since we are in the quantum setting we can compute O in polynomial time.…”

Applications of Algebra for Some Game Theoretic Problems

“…This is at present best-known upper bound due to [29]. [31] surveys results relating to the complexity of computing Galois groups and other related problems.…”

“…This gives worst case time O(K + · |G ′ | · |X|). If a univariate polynomial has n roots, then |G ′ | is linear in n [29,31], while |X| is polynomial in n. Finally, the verification of a Nash equilibrium solution is a polynomial time operation in the size of total number of strategies K + . This along with Proposition 20 gives us the following result:…”