AAED2

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Let $R\,$ be a Euclidean Domain with a function $\varphi$ . Prove that
(a) $\varphi(1) = \min\{\varphi(a) \ |\ a \in R \backslash \{0\}\}$

For all $a \in R \,\backslash \{0\}$ , the property of the function gives

$\varphi (1)$	$\leq \varphi (1\cdot a)$
$\varphi (1)$	$\leq \varphi (a)$

Therefore, $\varphi (1)$ has the minimum value.

(b) $R^\times = \{r \in R \backslash \{0\} \ |\ \varphi(r) = \varphi(1)\}$

Let $S = \{r \in R \,\backslash \{0\} \ | \ \varphi (r) = \varphi (1)\}$ .
( $R^\times \subseteq S$ ): Let $x \in R^\times$ . Then, $x^{-1} \in R^\times$ . So, by property of the function

$\varphi (x)$	$\leq \varphi (x\cdot x^{-1})$
$\varphi (x)$	$\leq \varphi (1)$

Since $\varphi (1)$ is the minimum value, $\varphi (x) = \varphi (1)$ . Therefore, $x \in S$ , and so $R^\times \subseteq S$ .

( $S \subseteq R^\times$ ): Let $x \in S$ . Then $\varphi (x) = \varphi (1)$ , and so $\varphi (x)$ has the minimum value. Now, since $R\,$ is a Euclidean Domain, there exists $q, r \in R$ such that $1 = qx + r\,$ , with $r = 0\,$ or $\varphi (r) < \varphi (x)$ . Since $\varphi (x)$ has the minimum value, $\varphi (r) < \varphi (x)$ is impossible, and so $r = 0\,$ . Thus, $1 = qx\,$ , and $x\,$ is a unit with inverse $q\,$ . Therefore, $x \in R^\times$ , and so $S \subseteq R^\times$ .

Therefore, $R^\times = \{r \in R \,\backslash \{0\} \ | \ \varphi (r) = \varphi (1)\}$ .

(c) Use (b) to determine $\mathbb{Z}^\times, F^\times, F[X]^\times,$ and $\mathbb{Z}[i]^\times$

For $\mathbb{Z}^\times$ , with $\varphi$ the absolute value, $\varphi (1) = |1| = 1$ , and so $\mathbb{Z}^\times = \{\pm 1\}$ .

For $F^\times$ , with $\varphi (x) = 1$ , $\varphi (1) = 1 = \varphi (x)$ for all $x \in F$ . Thus $F^\times = F \,\backslash \{0\}$ .

For $F[x]^\times$ , with $\varphi (p(x)) = \deg p(x)$ , $\varphi (1) = \deg 1 = 0$ , and so $F[x]^\times =$ non-zero polynomials of degree zero $= F \,\backslash \{0\}$ .

For $\mathbb{Z}[i]^\times$ , with $\varphi (a + bi) = a^2 + b^2$ , $\varphi (1) = 1^2 + 0^2 = 1$ , and so $\mathbb{Z}[i]^\times = \{\pm 1, \pm i\}$ .

Main Page : Abstract Algebra : Euclidean Domains

AAED2

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