臺灣大學數學系

八十九學年度第一學期碩博士班資格考試試題

 代數

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Do 6 out of the following 8 problems, including at least one from each set.

I.

1.

Show that a group of order 385 contains a normal subgroup of order 77.

2.

Let $p$ be a prime and $C_p$ the cyclic group of order $p$. How many subgroups of order $p$ are there in $G= C_p \oplus C_p \oplus C_p$? Construct another abelian group that is not isomorphic to $G$ but contains the same number of subgroups of order $p$ as $G$.

II.

1.

Let $R$ be the ring of all $n \times n$ matrices over a field $F$. Show that $R$ is a simple ring, that is, $R$ has no ideal other than $0$ or $R$.

2.

Let $R$ be a ring in which thet equation $ax=b$ is always solvable in $R$ for $a,b \in R$ with $a \neq 0$. Show that $R$ is a division ring if $R$ contains more than one elements.

III.

1.

Let $F$ be an infinite field and $K=F(x)$ where $x$ is an indeterminate over $F$. Show that $K$ is Galois over $F$, that is, the only subfield of $K$ that is fixed by all $F$-automorphisms of $K$ is $F$.

2.

Find the Galois group of $x^4+2$ over the field $\mathbb{Q}$ of rationals.

IV.

1.

Let $G$ be the group of all invertible $n \times n$ matrices over a field $F$. Show that any matrix in $G$ that commutes with all the matrices in $G$ must be a nonzero scalar matrix.

2.

Let $T$ be a linear transformation of a finite-dimensional vector space over a field of characteristic $0$. Show that $T^m =0$ for some integer $m$ if $T^i$ has trace $0$ for all $i \ge 1$.

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