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单词 ProofThatAGcdDomainIsIntegrallyClosed
释义

proof that a gcd domain is integrally closed


Proposition 1.

Every gcd domain is integrally closedMathworldPlanetmath.

Proof.

Let D be a gcd domain. For any a,bD, let GCD(a,b) be the collection of all gcd’s of a and b. For this proof, we need two facts:

  1. 1.

    GCD(ma,mb)=mGCD(a,b).

  2. 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 fractionMathworldPlanetmath of D, and a/bK (a,bD and b0) 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, ban. 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 associateMathworldPlanetmath of 1, hence a unit and we have a/bD.

Together with the additional property (call it property 3)

if GCD(a,b)=[1] and abc, then ac (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 cab with a,bD.Let r=gcd(a,c) and rt=a, rs=c. Then 1=gcd(s,t) by property (1) above. Next, since cab, write cd=ab so that rsd=rtb. This implies that sd=tb. So stb together with gcd(s,t)=1 show that sb by property (3). So we have just shown the existence of r,sD with c=rs, ra and sb. Therefore, c is primal and D is a Schreier domain.

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