Computational Class Field Theory

Abstract

Abstract. Class field theory furnishes an intrinsic description of the abelian extensions of a number field that is in many cases not of an immediate algorithmic nature. We outline the algorithms available for the explicit computation of such extensions.

Large Selmer groups over number fields

“…We will write down the elliptic curves E over Q in Legendre normal form and exhibit the number fields F in terms of class field theory. Defining polynomials can then be computed using algorithms described in [4,Chapter 6].…”

“…We will follow the notation in [4] so that the construction is readily implementable on a computer using the algorithms described there. Notation.…”

Topics in computational algebraic number theory

“…It is easy to compute p n , n ∈ Z, from the above data; see [10,Proposition 2.3.15], which treats the case n 0. The general case is an exercise: let p = (p, π) and n 0 = |n| /e(p/p) .…”

“…This description has a computational counterpart via Kummer theory, developed in particular by Cohen [10] and Fieker [17], relying heavily on efficient computation of the groups Cl f (K) in the following sense: Definition 1.1. a finite abelian group G is known algorithmically when its Smith Normal Form (SNF)…”

“…Acknowledgements : It is hard to overestimate what we owe to Henri Cohen's books [9,10], the state-of-the-art references on the subject. We shall constantly refer to them, supplying implementation details and algorithmic improvements as we go along.…”