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\centerline{\bf ALGEBRA PRELIMINARY EXAM}

\centerline{\bf JANUARY 18, 2000}

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\item{1.} Show that a Sylow $p$-subgroup of $D_{2n}$, the dihedral group of order $2n$, is cyclic and normal for every odd prime $p$.

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\item{2.} Show that every automorphism of $S_3$, the symmetric group of order $6$, is inner, that is, is conjugation by an element of $S_3$.

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\item{3.} Determine all abelian groups of order $144$ up to isomorphism.

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\item{4.} Let $f(x) = x^8 - 16$. Find the Galois group of the splitting field of $f(x)$ over the field of :

(a) rational numbers ;

(b) real numbers ;

(c) integers modulo $17$.

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\item{5.} Find two non isomorphic rings of $9$ elements whose additive groups are isomorphic. You must show that these rings are not isomorphic and that their additive groups are isomorphic.

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\item{6.} (a) Let $A$ be a commutative ring. Prove that every maximal ideal of $A$ is a prime ideal of $A$.   

\itemitem{(b)} Show that the ideal $(3, X)$ of ${\bold{Z}}[X]$ generated by $3$ and $X$ is a maximal ideal of ${\bold{Z}}[X]$.

\itemitem{(c)} Find a prime ideal of ${\bold{Z}}[X]$ that is not maximal.

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\item{7.} Let ${\bold{Z}}[i]$ denote the ring of Gaussian integers. Prove or disprove: ${\bold{Z}}[i] \otimes_{\bold{Z}} {\bold{R}} \cong {\bold{C}}$ as rings.

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\item{8.} (a) Prove or disprove : The ${\bold{Z}}$-module ${\bold{Z}}/2{\bold{Z}}$ is projective.

\itemitem{(b)} Prove or disprove : The ${\bold{Z}}$-module ${\bold{Z}}$ is flat.

\item\item{(c)} Prove or disprove : The ${\bold{Z}}$-module ${\bold{Q}}$ is injective.

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