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\centerline{\bf Algebra Preliminary Exam}

\centerline{\bf January 1997}

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\item{1.} Prove that if $A$ is an $n\times n$ matrix with coefficients in a field, then $A$ is similar to a matrix of the form $\left( \matrix
A_1 & & & & & 0 \\
& A_2 \\
& & . \\
& & & . \\
& & &  & . \\
0 & & & & & A_r
\endmatrix
\right)$ where the characteristic polynomial of $A_i \ (i=1 \ \dots \ r)$ is the power of an irreducible polynomial.

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\item{2.} Prove that $(p-1)! \equiv -1$ (mod $p$) for $p$ an odd prime.

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\item{3.} If $G$ is any group and $H$ is a subgroup of $G$ with $G:H=n$, show that there exists a normal subgroup $K$ of $G$ such that $K\subseteq H$ and $G:K\leq n!$

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\item{4.} Determine the structure of the Galois group $G$ of the splitting field $M$ over the rational numbers $Q$ of the polynomial $f(x) = x^5-2$.  How many Sylow 2-subgroups does $G$ have?  Give the fixed subfields of $M$ of each Sylow 2-subgroup.  Do the same thing for the Sylow 5-subgroups.  Which of these subfields are normal field extensions of $Q$?

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\item{5.} Let $K$ be a normal, separable extension field of $F$, and $p(x) \in F[x]$ be an irreducible polynomial.  If in $K[x] \ p(x) = p_1(x) \cdot \ \dots \ \cdot p_r(x)$ where $p_i(x)$ are irreducible polynomials in $K[x]$, $i=1 \dots r$, prove that $p_1(x),\dots,p_r(x)$ all have the same degree.

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\item{6.} Let $R$ be an integral domain.  State and prove the universal mapping property for the embedding of $R$ into its field of fractions.

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\item{7.} Let $R$ be a ring with unit, $A$, $C$ right $R$-modules, $B$, $D$ left $R$-modules, $f: A\to C$ a right $R$-module homomorphism, $g: B\to D$ a left $R$-module homomorphism.  Let $h: A \otimes_R B \to C \otimes_R D$ be defined by $h(a\otimes b) = f(a) \otimes g(b)$.  If $f$ and $g$ are monomorphisms, is $h$ necessarily a monomorphism?  Why?

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\item{8.} Let $p$ be a prime number and ${\bold{Z}}_p$ be the completion of ${\bold{Z}}$ at the prime ideal $p{\bold{Z}}$.  Prove that there exists a map
$$\chi: {\bold{F}}_p \to {\bold{Z}}_p$$
with the following properties:

(a) If $\pi: {\bold{Z}}_p \to {\bold{F}}_p$ is the canonical map, then
$$\pi \cdot \chi \ \text{is the identity on} \ {\bold{F}}_p$$

(b) $\chi$ is multiplicative:  that is, $\chi(ab) = \chi(a) \ \chi(b)$ for all $a,b$ in ${\bold{F}}_p$.


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