proof of the ring of integers of a number field is finitely generated over ℤ
Proof:
Choose any basis α1,…,αn of K over ℚ. Using the theorem in the entry multiples of an algebraic number
, we can multiply each element of the basis by an integer to get a new basis α1,…,αn with each αi∈𝒪K.
Consider the group homomorphism
φ:K→ℚn:γ↦(TrKℚ(γα1),…,TrKℚ(γαn)) |
where TrKℚ is the trace (http://planetmath.org/trace2) from K to ℚ. Note that φ is 1-1, since if γ≠0 and φ(γ)=0, then
n=TrKℚ(1)=TrKℚ(γγ-1)=TrKℚ(γ∑riαi)=∑riTrKℚ(γαi)=0 |
where the last equality holds since γ∈kerφ.
Hence φ:𝒪K↪ℤn, so 𝒪K is finitely generated and torsion-free. It has rank ≥n since the αi are linearly independent
, and rank ≤n since it injects into ℤn, so it has rank n.
Title | proof of the ring of integers of a number field is finitely generated over ℤ |
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Canonical name | ProofOfTheRingOfIntegersOfANumberFieldIsFinitelyGeneratedOvermathbbZ |
Date of creation | 2013-03-22 16:03:07 |
Last modified on | 2013-03-22 16:03:07 |
Owner | rm50 (10146) |
Last modified by | rm50 (10146) |
Numerical id | 9 |
Author | rm50 (10146) |
Entry type | Proof |
Classification | msc 13B22 |