Cyclotomic Fields and Class Group

Stream: new members

[Dev-Indra]

This is unrelated to Zulip, but does anyone know a book, article, pdf, that proves why the ideal class group of cyclotomic integers in finite? and whether prime ideals in this ring have prime norm?

[Kevin Buzzard]

The ideal class group of any global field is finite, and for any number field other than the rationals there are prime ideals with non-prime norm.

[Dev-Indra]

Why is the first claim true? (I know its true for quadratic number fields only)

[Kevin Buzzard]

This is a standard fact proved in any book on algebraic number theory at this level e.g. Marcus "Number fields".