\input vanilla.sty \magnification=1200 \baselineskip=10pt \nopagenumbers \centerline{\bf{Department of Mathematics and Statistics}} \bigskip \centerline{\bf{Preliminary Ph.D. Examination in Algebra}} \bigskip \centerline{\bf{January 18, 1994}} \midspace{.15in} \item{1.} (a) Let $G$ be a non-abelian group of order 8. Prove that there exists a cyclic subgroup \hskip15pt$H$ of order 4 in $G$. Determine whether this subgroup $H$ is normal in $G$. \bigskip \item\item{(b)} Give an example of a group of order 8 which does not admit any cyclic subgroup \hskip15pt of order 4. \bigskip \item{2.} Let $K$ be a finite field of cardinality $q$. \bigskip \item\item{(a)} Show that $K$ has characterstic $p$ for some prime $p$. \bigskip \item\item{(b)} Show that the group of ring automorphisms $\text{Aut}(K)$ is cyclic of order $\log_p q$. \bigskip \item{3.} Let $A_4$ denote the group of ``even'' permutations of a set of cardinality 4. \bigskip \item\item{(a)} Determine the 2-Sylow subgroup(s) of $A_4$. \bigskip \item\item{(b)} Let $T$ be a 3-Sylow subgroup of $A_4$. Show that $T$ is not normal in $A_4$. \bigskip \item\item{(c)} Show that any group of order 12 which does not have a normal 3-Sylow subgroup \hskip15pt is isomorhic to $A_4$. \bigskip \item{4.} Let ${\bold{Q}}$ denote the field of rational numbers. Let $K$ denote the field obtained by adjoining all the complex third roots of 2 to ${\bold{Q}}$. \bigskip \item\item{(a)} Determine the degree $[K : {\bold{Q}}]$. \bigskip \item\item{(b)} Determine the Galois group of the extension $K/{\bold{Q}}$. \bigskip \item\item{(c)} Determine all the subfields of $K$. \newpage \item{5.} Let $A$ be a ring and $0\to K\to F{\buildrel \pi \over \rightarrow} M \to 0$ , $0 \to K'\to F'{\buildrel \pi' \over \rightarrow} M \to 0$ be two short exact sequences of left $A$-modules, where $F$ and $F'$ are free $A$-modules. Show that $F\oplus K' \cong F' \oplus K$ as left $A$-modules as follows: \bigskip Let $Y = \{ (f, f')\in F \oplus F'|\pi(f) = \pi'(f')\}$. \bigskip Let $g : Y\to F$ by $g(f,f') = f$ and let $g' : Y \to F'$ by $g'(f,f') = f'$. \bigskip Show that \bigskip \item\item{(i)} $g$ and $g'$ are surjective. \bigskip \item\item{(ii)} $Y \cong F \oplus \text{ker}(g)\cong F'\oplus \text{ker}(g')$. \bigskip \item\item{(iii)} $\text{ker}(g) \cong K'$ and $\text{ker}(g') \cong K$. \bigskip Hence $F \oplus K' \cong F' \oplus K$. \bigskip \item{6.} Let $R$ be a commutative ring. \bigskip \item\item{(a)} Show that $P$ is a projective $R$-module iff $\text{Hom}(P,-)$ when applied to a short exact \hskip15pt sequence of $R$-modules is exact. \bigskip \item\item{(b)} Prove that every free module is projective. \bigskip \item\item{(c)} Give an example of a projective which is not free. \bigskip \item\item{(d)} Classify all projective modules over principal ideal domains. \bigskip \item{7.} (a) Show that every commutative ring with identity has a maximal ideal. \bigskip \item\item{(b)} Give an example of a ring with a unique maximal nonzero ideal. \bigskip \item\item{(c)} Give an example of a ring with a finite $( > 1)$ number of maximal ideals. \bigskip \item\item{(d)} Give an example of a ring with an infinite number of maximal ideals. \bigskip \item{8.} Let $M_n(F)$ denote the ring of n-by-n matrices with entries in a field $F$. \bigskip \item{ } Let $E(i,j)$ denote the matrix with $(i,j)$-entry equal to 1 and all other entries equal to 0. Let $A$ be a matrix in $M_n(F)$. \bigskip \item\item{(a)} Describe the result of multiplication of $A$ by $E(i,j)$ on the left and on the right \hskip15pt respectively. \bigskip \item\item{(b)} Show that $M_n(F)$ has no non-trivial proper two-sided ideals. \bye