Weston-idelic

# Weston-idelic - THE IDELIC APPROACH TO NUMBER THEORY TOM...

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Unformatted text preview: THE IDELIC APPROACH TO NUMBER THEORY TOM WESTON 1. Introduction In classical algebraic number theory one embeds a number field into the cartesian product of its completions at its archimedean absolute values. This embedding is very useful in the proofs of several fundamental theorems. However, it was noticed by Chevalley and Weil that the situation was improved somewhat if the number field is embedded in the cartesian product of its completions at all of its absolute values. With a few additional restrictions, these objects are known as the adeles , and the units of this ring are called the ideles. When considering the adeles and ideles, it is their topology as much as their algebraic structure that is of interest. Many important results in number theory translate into simple statements about the topologies of the adeles and ideles. For example, the finiteness of the ideal class group and the Dirichlet unit theorem are equivalent to a certain quotient of the ideles being compact and discrete. We will begin by reviewing the construction of local fields, first algebraically and then topologically. We will then prove the basic global results combining all of the local data, namely the product formula and the approximation theorem. Next we will define the adeles and the ideles and prove their basic topological properties. We will then define the idele class group, and relate it to the usual ideal class group. We will conclude with proofs of the finiteness of the ideal class group and the Dirichlet unit theorem, using idelic methods. I have tried in this paper to emphasize the topological details in these construc- tions, and hopefully have not ignored any important points. We will assume some familiarity with number fields, at the level of [3, Chapter 1], [4, Chapters 1-3] or [8, Chapter 1]. We fix the following notation throughout this paper: we let k be a number field (that is, a finite extension of Q ) of degree n over Q . We let o k be the ring of integers of k . If p is a prime ideal of o k with p ∩ Z = ( p ), we write e p for the ramification degree and f p for the inertial degree of p over p . That is, e p is the largest power of p dividing p o k , and f p is the degree of the residue field extension o k / p over Z / ( p ). If the prime p is clear from context, then we will just write e = e p and f = f p . Part 1. Classical Algebraic Number Theory 2. Local Fields : Algebraic Description Recall that the local ring o p ⊆ k is a discrete valuation ring. Let π be a uni- formizing element of o p ; that is, π generates the unique non-zero prime ideal po p of o p . Then any α ∈ k * can be written as α = uπ m for a unique integer m and some unit u of o * p . We say that this m is the p-adic valuation v p ( α ) of α . Setting 1 2 TOM WESTON v p (0) = ∞ , we have defined a discrete valuation v = v p : k → Z ∪ {∞} , which is easily checked to satisfy the usual properties of a discrete valuation: (1) v p ( αβ ) = v p ( α ) + v p ( β...
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Weston-idelic - THE IDELIC APPROACH TO NUMBER THEORY TOM...

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