Ring of integers of a number field

Ideal class group of a number field

Let be the ring of integers of a number field , where by abuse of terminology we refer to the ideal class group as the ideal class group of . ring The size of , which by ^P2 is finite, is called the class number.

Properties

1. Every ideal class contains a nonzero ideal of norm at most , Minkowski’s bound.
2. is finite.

Proof of 1–2

Let and so that . By Minkowski’s bound, there exists an so that

Then so since we are in a Containment-division ring, for some ideal , whence . By multiplicativity of the norm, , proving ^P1.

^P2 follows directly from ^P1 and ^P3.

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