Computational Class Field Theory
In Chapter 3 we gave the main theoretical results concerning global class field theory over number fields. We are now going to study this subject from the algorithmic point of view. In the present chapter, we give efficient algorithms for computing ray cl
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In Chapter 3 we gave the main theoretical results concerning global class field theory over number fields. We are now going to study this subject from the algorithmic point of view. In the present chapter, we give efficient algorithms for computing ray class groups of number fields and for computing the conductor and norm group of the Abelian extensions corresponding to congruence subgroups of these ray class groups by Takagi's Theorem 3.5.1. Thanks to Proposition 3.5.8 and Theorem 3.5.11, this allows us to compute their signature and discriminant. In the next two chapters, we will explain how to solve the more difficult problem of explicitly constructing relative or absolute defining polynomials for these Abelian extensions, and we will give some applications, particularly to the construction of number fields of small discriminant. The following exact sequence associated to the ray class group corresponding to a modulus m is an immediate consequence of Proposition 3.2.3: 1 ---+ (ZK/m)* /Im(U(K)) ---+ Clm(K) ---+ Cl(K) ---+ 1 .
To compute the ray class group Clm(K) from this exact sequence, there are three problems that a priori may seem difficult. First, the exact sequence may not split. Hence, although it may be easy to compute the cardinality h m of Clm(K), it may not be easy to compute its structure. Second, we will need to compute the structure of the group (ZK/m)*, and again this may not be simple. Finally, we need to compute the image of the units in this group and compute the quotient. When doing this by hand, one gets the impression that these tasks are not easy. In fact, this is quite a false impression, and we are going to see that suitable systematic use of the (ordinary) Smith and Hermite normal forms will lead to a nice and complete algorithmic solution to all of the above problems. Using the same tools, we can also compute the group Um(K), which also enters in Proposition 3.2.3 (see Exercise 1). Thus, the basic tools we will need are algorithms to compute with Abelian groups. This chapter is divided as follows. In Section 4.1, we describe the tools necessary for dealing with finitely generated Abelian groups (usually finite). In Section 4.2, we apply these tools to the algorithmic computation of the groups (ZK/m)* for an arbitrary modulus m. In Section 4.3, we give a complete algorithm for computing ray class groups of number fields and give the H. Cohen, Advanced Topics in Computational Number Theory © Springer Science+Business Media New York 2000
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4. Computational Class Field Theory
corresponding principal ideal algorithms. In Section 4.4, we explain how to perform a number of additional explicit computations in class field theory. We defer to Chapters 5 and 6 for algorithms to compute explicit polynomials and for examples.
4.1 Algorithms on Finite Abelian
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4.1.1 Algorithmic Representation of Groups In this section, which is an expanded version of [Co-Di-OI7], we consider finitely generated (in fact, usually finite) Abelian groups, which in view of our applications will be written
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