4.4 Factorization Theory in Dedekind Domains and the Fundamental Theorem of Algebraic Number Theory

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For the first three lemmas, assume that R is an arbitrary Dedekind domain.

Lemma 4.4.1 Assume that P1, P2, · · · , Pr, P are prime ideals in R with

P1P2 · · · Pr _ P.

Then P = Pi for some i.

Lemma 4.4.2 Any ideal of R contains a product of prime ideals.

Definition. Let R be a Dedekind domain. If I _ R is an ideal, we set

I−1 = {_ 2 E| _ · I _ R}.

Note that R−1 = R, for if _ ·R _ R, then _ = _ · 1 2 R. Next note that

I _ J implies that I−1 _ J−1.

Lemma 4.4.3 If I is a proper ideal of R, then I−1 properly contains R.

Lemma 4.4.4 If I _ R is an ideal then I−1 is a finitely generated R-module.

Proposition 4.4.5 If I _ R is an ideal, then I−1I = R.

Corollary 4.4.5.1 If I, J _ R are ideals, then (IJ)−1 = I−1J−1.

The following theorem gives us basic factorization theory in a Dedekind

domain.

Theorem 4.4.6 Let R be a Dedekind domain and let I _ R be an ideal.

Then there exist prime ideals P1, P2, · · · , Pr _ R such that

I = P1P2 · · · Pr.

The above factorization is unique in that if also

I = Q1Q2 · · ·Qs,

where the Qi’s are prime ideals, then r = s and Qi = P_(i), for some

permutation _ of 1, 2, · · · , r.

98 CHAPTER 4. DEDEKIND DOMAINS

The following theorem sometimes is called the Fundamental Theorem of

Algebraic Number Theory.

Corollary 4.4.6.1 (Fundamental Theorem of Algebraic Number Theory)

Let E _ Q be a finite field extension and let R = OE. Then any ideal of R

can be uniquely factored as a product of prime ideals.

From the Fundamental Theorem of Algebraic Number Theory, we conclude

that if I, J _ R are ideals that share no prime ideal factors, then it

must happen that I + J = R, i.e., the ideals I, J are relatively prime. In

particular let I _ R be an ideal and and factor I into a product of distinct

prime ideals: I = Pe1

1 Pe2

2 · · · Per

r . Let _i 2 Pei

i − Pei+1

i+1 , i = 1, 2 . . . , r.

Since Pe1

1 , Pe2

2 . . . , Per

r are pairwise relatively prime, by the Chinese Remainder

Theorem (see Exercise 7 of Section 3.1) there exists an element

_ 2 R satisfying _ _= _i mod Pei+1

i+1 , i = 1, 2, . . . ,R. Note that in particular

_ 2 Pe1

1 \ Pe2

2 \ · · · \ Per

r = Pe1

1 Pe2

2 · · · Per

r (see Exercise 2, below).

This implies that if we factor the principal ideal (_) into a product of prime

ideals, then we have (_) = Pe1

1 Pe2

2 · · · Per

r · J where J is divisible by none

of the prime ideals P1, P2, . . . , Pr. In other words, we have a factorization

(_) = IJ, where I, J are relatively prime.

Next, write J = Qf1

1 Qf2

2 · · ·Qfs

s ; from the above we may infer that I 6_

Qi, i = 1, 2, . . . , s, and so by Exercise 9 of Section 3.1 we may conclude that

I 6_ Q1 [Q2 [· · ·[Qs. Now choose an element _ 2 I −(Q1 [Q2 [· · ·[Qs).

Therefore the ideal (_, _) _ R generated by _ and _ satisfies (_) _ (_, _) _

I. However, since (_, _) 6_ Qi, i = 1, 2 =, . . . , s, we may infer that in fact,

(_, _) = I. This proves the following:

Proposition 4.4.7 Let E _ Q be a finite field extension and let R = OE.

Then any ideal I _ R can be expressed as I = (_, _) for suitable elements

_, _ 2 I.

Exercises

1. Let E be a finite extension of the rational field Q, and set R = OE. Let

P be a prime ideal of R, and assume that P \Z = (p), for some prime

number p. Show that we may regard Z/(p) as a subfield of R/P, and

4.4. FACTORIZATION THEORY IN DEDEKIND DOMAINS 99

that [R/P : Z/(p)] _ [E : Q], with equality if and only if p remains

prime in OE.

2. Assume that R is a Dedekind domain and that I = Pe1

1 Pe2

2 · · · Per

r ,

J = Pf1

1 Pf2

2 · · · Pfr

r . Show that

I+J = P

min{e1,f1}

1 · · · Pmin{er,fr}

r , I\J = P

max{e1,f1}

1 · · · Pmax{er,fr}

r .

Conclude that AB = (A + B)(A \ B).

3. Let R be a Dedekind domain in which every prime ideal is principal.

Prove that R is a p.i.d.

4. In the Dedekind domain R = Z[

p

−5] show that (3) = (3, 4+

p

p −5)(3, 4−

−5) is the factorization of the principal ideal (3) into a product of

prime ideals.

100 CHAPTER 4. DEDEKIND DOMAINS

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