臺灣大學數學系

八十六學年度第一學期碩博士班資格考試試題

代數

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1. Prove that $\mathbb{Q}$ has no proper subgroups of finite index. Deduce that $\mathbb{Q}/\mathbb{Z}$ has no proper subgroups of finite index, where $\mathbb{Q}$ is the additive group of rational numbers and $Z$ is the additive group of integers.
2. Let $G$ be a finite group of odd order and $x$ a nonidentity element in $G$. Prove that $x$ and $x^{-1}$ are not conjugate in $G$.
3. Let $R={\rm M}_n(F)$, the $n$ by $n$ matrix ring over a field $F$, $n>1$ and $a, b$ nonzero elements in $R$. Show the following statements:
   1. ${\rm rank}(a)={\rm rank}(b)$ if and only if $\dim_FaR=\dim_FbR$.
   2. $axa=a$ for some $x\in R$.
   3. $aR=eR$ for some element $e=e^2\in R$.
4. Prove that $\mathbb{Z}[\sqrt {-2}]=\{ a+b\sqrt {-2}\mid a, b \in \mathbb{Z}\}$ is a Euclidean domain with the norm $N(a+b\sqrt {-2})=a^2+2b^2$, where $\mathbb{Z}$ is the ring of integers. Also, find $q, r\in \mathbb{Z}[\sqrt {-2}]$ such that $35+23\sqrt {-2}= \big( 5+7\sqrt {-2}\big) q + r$ with $N(r)<123$.
5. Let $K$ be the splitting field of $X^4-7$ over $\mathbb{Q}$, the field of rational numbers. Find the Galois group $G$ of $K$ over $\mathbb{Q}$ and describe the correspondence between the subgroups of $G$ and the subfields of $K$.

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