Sintlattice

# Sintlattice - Math 676 The lattice of S-integers Let K be a...

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Unformatted text preview: Math 676. The lattice of S-integers Let K be a number field, and let S be a finite set of places of K containing the set S ∞ of archimedean places. Recall that we define the ring O K,S of S-integers in K to be the set of a ∈ K such that a is v-integral for all (necessarily non-archimedean!) v 6∈ S ; that is, || a || v ≤ 1 for all v 6∈ S . For a ∈ O K,S , we have a ∈ O × K,S if and only if a 6 = 0 and a, 1 /a ∈ K × each lie in O K,S . That is, a ∈ O × K,S if and only if a ∈ K × and || a || v , || 1 /a || v ≤ 1 for all v 6∈ S . This final condition says || a || v = 1 for all v 6∈ S . Hence, O × K,S = { x ∈ K × ||| x || v = 1 for all v 6∈ S } . 1. Preliminaries We first wish to show that O K,S can be concretely constructed from O K and knowledge of the class number. For each non-archimedean place of K , we let p v denote the corresponding prime ideal of O K . Since the class group is killed by the class number, for all non-archimedean v the ideal p h ( K ) v in O K is principal. Hence, the finite product Q v ∈ S- S ∞ p h ( K ) v has the form a S O K , so 1 /a S ∈ K × is non-integral at precisely those non-archimedean v that lie in S (if S = S ∞ then this product is empty and we may interpret the product over the empty set S- S ∞ to be the unit ideal O K ; a S is an element of O × K in this case). Having constructed one such element, we now show that any such element allows us to construct O K,S as a localization of O K : Lemma 1.1. For a ∈ O K- { } , we have O K,S = O K [1 /a ] if and only if the finite set of non-archimedean v for which || 1 /a || v > 1 is exactly the set S- S ∞ . ( Equivalently, the condition is that the prime factors of a O K are exactly the primes p v for v ∈ S- S ∞ ) . Proof. If O K,S = O K [1 /a ] then 1 /a is v-integral for all v 6∈ S , so || 1 /a || v ≤ 1 for all v 6∈ S . We wish to show that || 1 /a || v > 1 for the other non-archimedean places, namely those v ∈ S- S ∞ . Suppose otherwise, so || 1 /a || v ≤ 1 for some v ∈ S- S ∞ . That is, assume 1 /a is v-integral for some non-archimedean v ∈ S . Since all elements of O K are also v-integral, it follows that all elements of O K,S = O K [1 /a ] are v-integral. However, this is not true: by finiteness of the class group we have p h ( K ) v = a O K for some a ∈ O K- { } , and clearly 1 /a ∈ O K,S (since a is even a local unit at all places not in S ) yet 1 /a is not v-integral for the place v ∈ S (as || 1 /a || v > 1 due to the prime factorization of a O K )....
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## This note was uploaded on 10/26/2011 for the course MATH 8410 taught by Professor Staff during the Fall '11 term at UGA.

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Sintlattice - Math 676 The lattice of S-integers Let K be a...

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