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Math Help - Euclidean Domains

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    Euclidean Domains

    Let R be a Euclidean Domain. Let m be the minimum integer in the set of norms of nonzero elements of R. Prove that every nonzero element of norm m is a unit in R . Deduce that a nonzero element of norm zero (if such an element exists) is a unit.
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    Quote Originally Posted by dori1123 View Post
    Let R be a Euclidean Domain. Let m be the minimum integer in the set of norms of nonzero elements of R. Prove that every nonzero element of norm m is a unit in R . Deduce that a nonzero element of norm zero (if such an element exists) is a unit.
    A norm is a function n: R^{\times}\to \mathbb{N} such that for any a,b\in R, b\not = 0 we can write a = qb + r where: r\not = 0 and n(r) < n(b), or r=0. (There might be an additional property to the norm function but not all books define it).

    Say that n(x) = m where m = \min \{ n(r) : r\in \mathbb{R}^{\times}\}. This means we can write, 1 = qx + r where r=0 or else n(r) < n(x). But n(x) is minimal and so r=0. Thus, 1=qx which means that x|1. Thus, x is a unit by definition. It follows that if n(x) = 0 then since x has the smallest possible norm it means it has to be a unit.
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