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**jaacob****Member**- Registered: 2010-04-21
- Posts: 2

1.Let p_1,p_2 e Z[x]. Z[p_1,p_2] is subring og Z[x] generated with Z U {p_1,p_2}

are Z[x^2 - x^5, x^2 - 2x^5], Z[x^2+x^6,x^2+2x^6] unique factorization domain?

2.Prove that the ring Z[2 *sqrt(-1)]={a+2b* sqrt(-1), a,b e Z} is not principial ideal domain. Is it Euclidean domain?

I need help because I'm not good at this

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

1.Let p_1,p_2 e Z[x]. Z[p_1,p_2] is subring og Z[x] generated with Z U {p_1,p_2}

are Z[x^2 - x^5, x^2 - 2x^5], Z[x^2+x^6,x^2+2x^6] unique factorization domain?

Why the first line? It doesn't seem to have anything to with the question. To answer the question, it's best to try to simplify as much as possible. For example, if you have x^2 - x^5 and x^2 - 2x^5 both in your ring, then so is

So your ring becomes Z[x^2, x^5]. Is this a UFD?

2.Prove that the ring Z[2 *sqrt(-1)]={a+2b* sqrt(-1), a,b e Z} is not principial ideal domain. Is it Euclidean domain?

If (a, b) = (c), then c is the gcd of a and b. So try to find something that doesn't have a gcd. As for the second part of the question, remember that the Euclidean norm gives you an algorithm for finding the gcd.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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