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Rings of algebraic numbers and functions

✍ Scribed by Edward D. Davis

Publisher
John Wiley and Sons
Year
1965
Tongue
English
Weight
411 KB
Volume
29
Category
Article
ISSN
0025-584X

No coin nor oath required. For personal study only.

✦ Synopsis

Introduction.

That every integrally closed subring of the field of algebraic numbers is a ring of quotients of its subring of algebraic integers is a remark of 131. The purpose of the present note is to prove this assertion without the hypothesis of integral closure (Theorem A). The proof rests on a theorem concerning I-dimensional Noetherian domains [ 5 , 6, 71 and the finiteness of the "class group" of a subring of the ring of integers in ail algebraic number field. (Here "class group" means invertible fractional ideals mod principal fractional ideals.) The finiteness of the class group is proved in 5 1; Theorem A and Theorem B, the analogous result for subrings of a field of transcendence degree 1 over a finite field, are proved in 5 2. I n 8 3 we consider the case of subrings of a field of transcendence degree 1 over an arbitrmy ground field.

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