4.3 OE is a Dedekind Domain

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Definition. Let R be an integral domain. We say that R is a Dedekind

domain if

(a) R is Noetherian,

(b) Every prime ideal of R is maximal, and

(c) R is integrally closed.

Thus, it follows immediately that every p.i.d is a Dedekind domain.

For the remainder of this section, let [E : Q] < 1, and set R = OE.

Lemma 4.3.1

(a) There exists _ 2 R such that E = Q[_].

(b) If _ is as above and if R0 = Z[_], then there exists d 2 Z with d·R _ R0.

Proposition 4.3.2 R is Noetherian.

Proposition 4.3.3 Every prime ideal of R is maximal.

Corollary 4.3.3.1 R is a Dedekind domain.

Because of Exercise 4 of Section 3.3, we have the following result, promised

in Section 4.2.

Corollary 4.3.3.2 The algebraic integer domain R is a p.i.d. if and only

if R is a u.f.d..

4.4. FACTORIZATION THEORY IN DEDEKIND DOMAINS 97

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