Ideal class group of a number field

Let $O 𝐾$ be the ring of integers of a number field $𝐾$ , where by abuse of terminology we refer to the ideal class group $C l (𝐾) : = C l (O 𝐾)$ as the ideal class group of $𝐾$ . #m/def/ring The size of $C l (𝐾)$ , which by ^P2 is finite, is called the class number.

Properties

Every ideal class contains a nonzero ideal of norm at most $𝑀 𝐾$ , Minkowski's bound.
$C l (𝐾)$ is finite.

Proof of 1–2

Let $𝑐 \in C l (𝐾)$ and $𝔞 ⊴ O 𝐾$ so that $[𝔞] = 𝑐 - 1$ . By Minkowski's bound, there exists an $𝛼 \in 𝔞$ so that

| N (𝛼) | \leq 𝑀 𝐾 N (𝔞) .

Then $⟨ 𝛼 ⟩ \subseteq 𝔞$ so since we are in a Containment-division ring, $⟨ 𝛼 ⟩ = 𝔞 𝔟$ for some ideal $𝔟 ⊴ O 𝐾$ , whence $[𝔟] = 𝑐$ . By multiplicativity of the norm, $N (𝔟) = | N (𝛼) | / N (𝔞) \leq 𝑀 𝐾$ , proving ^P1.

^P2 follows directly from ^P1 and ^P3.

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