課程名稱︰代數二
課程性質︰數學系選修 數學系碩士班數學組必選
課程教師︰朱樺
開課系所︰數學系所
考試時間︰950506
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
(1) (30%) Let $R=\mathbb{Z}[x}$.
(a) Show that any ideal of $R$ is finitely generated.
(b) Let $M$ be a maximal ideal of $R$. Show that it is impossible that
$M \cap \mathbb{Z}=(0)$.
(c) Show that any maximal ideal of $R$ can be generated by two elements.
(d) Show that, for any $n$, there is an ideal cannot be generated by $n$
elements.
(2) (15%) Let $R$ be a commutative ring, $M$ be an $R$-module. Then the
annihilator of $M$ is $\Ann_R(M)=\{a \in R| aM=0\}$.
Let $P$ be a finitely generated prime ideal of $R$ with annihilator $(0)$.
Show that $\Ann_R(P/P^2)=P$.
(3) (20%) Let $R=\mathbb{C}[x,y,z]$. Find the intersection of ideals
$(xy+z, x+yz) \cap (x+z)$.
(4) (25%) Let $F=\mathbb{Q}(\sqrt[3]{2})$ and
$\alpha=a+b\sqrt[3]{2}+c\sqrt[3]{4} \in F$.
(a) Find a monic cubic polynomial $f(x) \in \mathbb{Q}[x]$ such that
$f(\alpha)=0$.
(b) Determine which elements $\alpha$ are algebraic integers.
(5) (25%) Let $R=\mathbb{Z}[\sqrt{-17}]$. Determine the ideal class group
of $R$. (\lfloor \mu \rfloor=2\sqrt{|D|}/\pi)
(6) (25%)
(a) Classfy finitely generated modules over the ring $R=F[x]/(f(x))$,
where $F$ is a field.
Let $F$ be any one of the following fields: $mathbb{R}$, $\mathbb{Q}$,
$\mathbb{F}_p$, $p$ is a prime, $f(x)=x^4+x^3+x^2+x+1$.
(b) Find all fields $F$ such that the additive group $F^1$ have a
structure of $R$-module.
(c) Find all fields $F$ such that the additive group $F^2$ have a
structure of $R$-module.
--
╓╮★╓╮ ╓╮
╟╢╰╫╫╮╟╢╓╖╮╮╭╥─╮╓╮ ╮╭╥─╮╭╥─╮
╟╢ ╟╢ ╟╢╟╢║║╟╢ ║╟╢ ║╰╨╥╮╟╢ ║
╟╢ ╟╢ ╟╢╟╢║║╟╢ ║╟╢ ║ ╟╢╟╫─╯
╰╜ ╰╨╯╰╜╰╜╜╜╰╨─╯╰╨─╰╰─╨╯╰╨─╯
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.50.48