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Math Help - Euclidean domains

  1. #1
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    Euclidean domains

    Hi,

    i need to show which of the following are euclidean domains:

    a) C[x] with d(f)=deg(f) (where C[x] = set of polynomials with complex coefficients)
    b) set of integers with d(n)=|n|+1
    c) ring of integers modulo 5 with d(f)=2deg(f)
    d) ring of integers modulo 6 with d(f)=deg(f)
    e) Z(w) where w=(1+root(-3))/2, with d(z)=|z|^2

    i know that an integral domain is a Euclidean domain if it has a Euclidean function but i dont know how to start off the question.

    Thanks!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by choo View Post
    Hi,

    i need to show which of the following are euclidean domains:

    a) C[x] with d(f)=deg(f) (where C[x] = set of polynomials with complex coefficients)
    b) set of integers with d(n)=|n|+1
    c) ring of integers modulo 5 with d(f)=2deg(f)
    d) ring of integers modulo 6 with d(f)=deg(f)
    e) Z(w) where w=(1+root(-3))/2, with d(z)=|z|^2

    i know that an integral domain is a Euclidean domain if it has a Euclidean function but i dont know how to start off the question.

    Thanks!
    these are all mostly plug and chug. I'll help with the first one.

    We may assume WLOG that p,p'\ne 0. Thus, \deg(pp')=\deg(p)+\deg(p')\geqslant deg(p).

    The other bit follows from this.

    Try the others and get back to us.
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  3. #3
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    for part b) this is what i got:

    the function is strictly increasing.
    a=bq+r
    |r| <= |b/2|
    this means that d(b/2) < d(b) ?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by choo View Post
    for part b) this is what i got:

    the function is strictly increasing.
    a=bq+r
    |r| <= |b/2|
    this means that d(b/2) < d(b) ?
    What does that matter? You want to prove that d(a)\leqslant d(ab) for every a,b\in \mathbb{Z}-\{0\}, right?

    So d(ab)=|ab|+1=|a||b|+1 now since b\in\mathbb{Z} we know that |b|\geqslant 1 and so |a||b|+1\geqslant |a|+1=d(a)
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  5. #5
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    thanks for ur help. heres what i did for the other parts:

    c) d(ab)=2deg(ab), b>=[1] and so
    2deg(ab)= 2(deg(a)+deg(b))>=2deg(a)=d(a)

    d) the ring of integers modulo 6 is not a field.
    d(ab)=deg(ab)=deg(a)+deg(b)>=deg(a)

    e) d(w)=|w|^2 = 1/4+3/4=1
    (what do i do next?)

    thanks
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