By Matthew Baker

Direction Notes (Fall 2006) Math 8803, Georgia Tech, model 24 Nov 2012

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**Extra info for Algebraic Number Theory [Course Notes]**

**Example text**

E, y ∈ p. Therefore x = γy ∈ γp as desired. 54. Note that it took a bit of messy verification to prove that OK /p ∼ = pk /pk+1 . In the case where p = (π) is principal, this is very simple: one proves easily in that case that multiplication by π k gives an isomorphism. In general, there is a useful technique called localization which allows one to prove certain results about Dedekind domains by first reducing them to the case of a PID. We will see this in action later on. Localization is also a very useful tool in other contexts.

It follows that pξ ∈ Z[α] and ξ ∈ / Z[α]. Write pξ = b0 + b1 α + · · · + bn−1 αn−1 . Then each bi ∈ Z, and not every bi is divisible by p. Let j be the smallest index with 0 ≤ j ≤ n − 1 for which p bj . Then bj−1 j−1 b0 b1 + α + ··· + α p p p bj bj+1 j+1 bn = αj + α + · · · + αn p p p η := ξ − ∈ OK , b since both ξ and bp0 + bp1 α + · · · + j−1 αj−1 are in OK . p If η ∈ OK , then we also have bj αn ηαn−j−1 = αn−1 + bj+1 + bj+2 α + · · · + bn αn−j−2 ∈ OK . p p Since αn = −(a0 + a1 α + · · · + an−1 αn−1 )/p ∈ OK , p b b it follows that pj αn−1 ∈ OK .

Therefore up ≡ a (mod p) also, and we have a ≡ −a (mod p). Since p is odd, this implies that a ≡ 0 (mod p), so that p | up . This contradicts the fact that u is a unit. 38. If p is an odd prime, then every unit in Z[ζp ] can be written as rζpj for some integer 0 ≤ j ≤ p − 1, where r is a real number belonging to the field Q(ζp ). Proof. Let u be a unit in Z[ζp ]. 37, we know that u/u = ζpk for some integer k. Choose an integer 0 ≤ j ≤ p − 1 such that 2j ≡ k (mod p). Then uζp−j = uζpj . , r ∈ R.

### Algebraic Number Theory [Course Notes] by Matthew Baker

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