proof that a gcd domain is integrally closed
Proposition 1.
Every gcd domain is integrally closed.
Proof.
Let D be a gcd domain. For any a,b∈D, let GCD(a,b) be the collection of all gcd’s of a and b. For this proof, we need two facts:
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1.
GCD(ma,mb)=mGCD(a,b).
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2.
If GCD(a,b)=[1] and GCD(a,c)=[1], then GCD(a,bc)=[1].
The proof of the two properties above can be found here (http://planetmath.org/PropertiesOfAGcdDomain). For convenience, we let gcd(a,b) be any one of the representatives in GCD(a,b).
Let K be the field of fraction of D, and a/b∈K (a,b∈D and b≠0) is a root of a monic polynomial p(x)∈D[x]. We may, from property (1) above, assume that gcd(a,b)=1.
Write
f(x)=xn+cn-1xn-1+⋯+c0. |
So we have
0=(a/b)n+cn-1(a/b)n-1+⋯+c0. |
Multiply the equation by bn then rearrange, and we get
-an=cn-1ban-1+⋯+c0bn=b(cn-1an-1+⋯+c0bn-1). |
Therefore, b∣an. Since gcd(a,b)=1, 1=gcd(an,b)=b, by repeated applications of property (2), and one application of property (1) above. Therefore b is an associate of 1, hence a unit and we have a/b∈D.
∎
Together with the additional property (call it property 3)
if GCD(a,b)=[1] and a∣bc, then a∣c (proof found here (http://planetmath.org/PropertiesOfAGcdDomain)),
we have the following
Proposition 2.
Every gcd domain is a Schreier domain.
Proof.
That a gcd domain is integrally closed is clear from the previous paragraph. We need to show that D is pre-Schreier, that is, every non-zero element is primal. Suppose c is non-zero in D, and c∣ab with a,b∈D. Let r=gcd(a,c) and rt=a, rs=c. Then 1=gcd(s,t) by property (1) above. Next, since c∣ab, write cd=ab so that rsd=rtb. This implies that sd=tb. So s∣tb together with gcd(s,t)=1 show that s∣b by property (3). So we have just shown the existence of r,s∈D with c=rs, r∣a and s∣b. Therefore, c is primal and D is a Schreier domain.
∎
Title | proof that a gcd domain is integrally closed |
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Canonical name | ProofThatAGcdDomainIsIntegrallyClosed |
Date of creation | 2013-03-22 18:19:27 |
Last modified on | 2013-03-22 18:19:27 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 6 |
Author | CWoo (3771) |
Entry type | Derivation |
Classification | msc 13G05 |