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\title{進階代數(上)第一次作業}
\date{2008~年九月十八日}
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\author{上課老師: 翁志文 }

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\pointname{分}

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%       這份試卷有 \   \numquestions\ 大題,\ 共計\  \numpoints\
%  分.\  請詳細提供所有過程!}}}
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We always assume that a ring has the multiplication identity $1.$

\bigskip

\begin{questions}
\question Let $Q$ denote the ring of real quarternions. For $x=a+bi+cj+dk\in Q$ the {\it conjugate}
of $x$ is $x^*:=a-bi-cj-dk.$
\begin{parts}
\part (連敏筠)  Show $(a+bi+cj+dk)(a-bi-cj-dk)=a^2+b^2+c^2+d^2$ for $a, b, c, d\in \mathbb{R}.$
\part (施智懷) Suppose $a_1, b_1, c_1, d_1, a_2, b_2,
c_2, d_2\in \mathbb{Z}.$ Show that there exist $a, b, c, d\in \mathbb{Z}$ such that
$$(a_1^2+b_1^2+c_1^2+d_1^2)(a_2^2+b_2^2+c_2^2+d_2^2)=a^2+b^2+c^2+d^2.$$
\part (邱鈺傑) Suppose $u\in \mathbb{Z}$ and
$2u=a^2+b^2+c^2+d^2$ for some $a, b, c, d\in \mathbb{Z}.$ Then $u=e^2+f^2+g^2+h^2$ for some $e, f,
g, h\in \mathbb{Z}.$ (Hint. Try $e=(a+b)/2$ and $f=(a-b)/2.$)
\end{parts}

設計理論會用到一個不太容易證的定理: 任意正整數都能寫成四整數的平分和.~ 你能猜測此定理證明的方向嗎?


\question Let $R$ be a commutative ring of prime characteristic $p.$
\begin{parts}
\part (蕭雯華) Show that
$$(a+b)^{p^n}=a^{p^n}+b^{p^n}$$ for all $a, b\in \mathbb{N}.$
\part (斐若宇) Show that the map $f:R\rightarrow R$ given by $f(a)=a^p$ is a homomorphism of rings.
\end{parts}


\question (林逸軒) An element $a$ of a ring is {\it nilpotent} if $a^n=0$ for some $n.$ Prove that
in a commutative ring $a+b$ is nilpotent if $a$ and $b$ are. Show that this result may be false if
$R$ is not commutative.


\question (陳巧玲 ) In a ring $R$ show that the following conditions are equivalent.
\begin{parts}
\part $R$ has no nonzero nilpotent elements.
\part If $a\in R$ and $a^2=0$, then $a=0.$
\end{parts}





\question (李光祥) Give an example of a nonzero homomorphism $f:R\rightarrow R'$ of rings such that
$f(1)\not=1'.$ Is it possible $1'$ in the image of $f$?


\question (林詒琪) Find a nonidentity isomorphism $\phi$ of $\mathbb{R}$ into $\mathbb{R}.$

\question (葉彬) Show that the only ring homomorphism $\phi$ of $\mathbb{R}$ into $\mathbb{R}$ with
$\phi(1)=1$ is the identity.
\end{questions}

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