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Math Help - Rings.

  1. #1
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    Rings.

    1.Let p_1,p_2 e Z[x]. Z[p_1,p_2] is subring of 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?

    Please help me.. i'm not good at this, but i need it..
    Last edited by mr fantastic; April 23rd 2010 at 03:45 PM. Reason: Re-titled post.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by tom007 View Post
    1.Let p_1,p_2 e Z[x]. Z[p_1,p_2] is subring of 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?

    Please help me.. i'm not good at this, but i need it..
    I mean, you've shown zero work. We have no idea where you are at in your skill levels?

    Can you answer these questions:

    What does generate mean?

    What is a unique factorization domain (UFD)?

    What is a principle ideal domain (PID)?

    What is a Euclidean Domain? (or maybe what is a Euclidean valuation?)
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  3. #3
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    An integral domain R is a unique factorization domain provided that:
    (i) every nonzero nonunit element a of K can he written a = C1C2...Cn, with
    C1, . . . , Cn irreducible.
    (ii) if a = C1C2.... Cn and a = d1d2....dm (Ci,di irreducible), then n = m and for
    some permutation o of { 1,2, . . . , n}, Ci and d(o(i)) are associates for every i.


    An ideal (x) generated by a single element is called a principal ideal. A principal ideal ring is a ring in which every ideal is principal. A principal ideal ring which is an integral domain is called a principal ideal domain.


    R is a Euclidean ring if there is a function v : R {0) -> N such that:
    (i) if a,b e R and ab=! 0, then v(a) <=v(ab);
    (ii) if a,b e R and b=! 0, then there exist q,r s R such that a = qb + r with r= 0, or r=!0 and v(r) < v(b).
    A Euclidean ring which is an integral domain is called a Euclidean domain.
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