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Thread: Euclidean Domains

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    Euclidean Domains

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

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