\documentclass[12pt]{article} \usepackage{amsfonts} \usepackage{amsmath, amsthm, amssymb} \usepackage{myfullpage} %%%% If you want 1 inch margins \pagestyle{myheadings} %------------------------------------------- % Begin Topmatter infomation %------------------------------------------- \title {Sample Algebra Questions} \author {Thomas R. Shemanske} \date{Last updated: January 2008} %\predate %With time and preliminary version appended \markboth{Sample Algebra Questions} {Sample Algebra Questions} %------------------------------------------- % End Topmatter infomation %------------------------------------------- %------------------------------------------- % Begin Local Macros %------------------------------------------- \newcommand{\Z}{{\mathbb Z}} \newcommand{\Q}{{\mathbb Q}} \newcommand{\C}{{\mathbb C}} \newcommand{\normal}{\triangleleft} %------------------------------------------- % End Local Macros %------------------------------------------- \emergencystretch=20pt \begin{document} \maketitle % ------------------------------------------- Other Topmatter % Information ------------------------------------------- % \dedication{This is dedicated to the one I love} % \subjclass {Primary 11Fxx; Secondary 11Fxx} % \keywords{Maximal Order, Central Simple Algebra, Bruhat--Tits % Building} % \begin{abstract} % This is a great paper. Read no further, because I don't want you % to hurt yourself, but if you can't help yourself, better strap in. % It gets bumpy from here on in. % \end{abstract} % ------------------------------------------- End Other Topmatter % Information ------------------------------------------- % \def\today{\ifcase\month\or January\or February\or March\or April\or % May\or June\or July\or August\or September\or October\or % November\or December\fi\space\number\day, \number\year} \section{Introduction} Graduate students often ask what they should study in preparation for a qualifying exam. The answer is of course that definitions, statements of theorems, examples and sketches of proofs of major theorems are the norm in a qualifying exam. However, this information is often not perceived by the student in the same way as it is intended by the faculty member. To address this disparity, I have included below a number of sample questions to help you prepare for the algebra qualifying exam. These questions reflect only some of my own prejudices, and are not meant to reflect questions which other examiners might ask. These questions are in no sense intended to reflect a comprehensive review of the material, but should give you a good idea of the type and depth of question which you might be expected to answer. In particular, if you are not comfortable with the vast majority of the material reflected by these questions, you are not ready to take the qual. In the exam itself, expect to be asked questions like those below, but also some which go beyond the bounds of your knowledge, and possibly outside the bounds of the syllabus. It is by probing the boundaries of your knowledge that we ascertain the depth of your knowledge. You are not expected to know the answer to everything we ask, but you are expected to know a majority. \section{Group Theory} \begin{enumerate} \item What is meant by a group acting on a set? Give two characterizations and show that they are equivalent. Give two examples of a group action. \item If a group $G$ acts on a set $S$, define the notion of an isotropy subgroup and of an orbit. How are these notions related when the group and/or set are infinite? When both the group and set are finite, what more can be said? \item Let $G$ be a finite group and let $H$ be a subgroup whose index in $G$ is the smallest prime dividing the order of the group $G$. Show that $H$ is a normal subgroup of $G$. \item Derive the class equation. Use it to prove that the center of a (nontrivial) $p$-group is nontrivial. \item Show that if $Z$ is the center of a group $G$, and $G/Z$ is cyclic, then $G$ is abelian. \item Show that if $p$ is a prime, any group of order $p^2$ is abelian. \item Let $G$ be a group and $G'$ its commutator subgroup, that is $G'$ is the subgroup of $G$ generated by elements of the form $xyx^{-1}y^{-1}$ with $x,y \in G$. Show that \begin{enumerate} \item $G' \normal G$ \item $G/G'$ is abelian \item If $H \normal G$ and $G/H$ is abelian, then $G' \subset H$. \end{enumerate} \item Show that if $G$ is a group and $N \normal G$, then $G$ is solvable if and only if $N$ and $G/N$ are solvable. \item Show that the following two statements are equivalent: \begin{enumerate} \item Every finite group of odd order is solvable. \item Every finite nonabelian simple group has even order. \end{enumerate} \item Suppose that $p$ is a prime, and that $G$ is a nonabelian group of order $p^3$. Let $Z$ denote the center of $G$. Show that \begin{enumerate} \item $G' = Z$, and \item $G/Z$ is the direct product of two cyclic groups of order $p$. \end{enumerate} \item Show that a finite abelian group has a subgroup of every order dividing the order of the group. \item Show that a finite group always has a composition series. \item Suppose that an arbitrary group $G$ has two subgroups $H$ and $K$ each of finite index in $G$. Show that the intersection $H \cap K$ has finite index in $G$; in particular, show that $[G:H\cap K] \leq [G:H] [G:K]$. \item Let $p < q$ be primes, and let $G$ be a group of order $pq$. Show that if $p \nmid (q-1)$ then $G$ is cyclic. \item Let $G$ be a finite group, and $p$ a prime dividing the order of $G$. Let $H= \{x\in G | o(x) = p^m,\ {\rm some\ } m\}$. Show that $H$ is the union of all the $p$-Sylow subgroups of $G$. \item Show that a finite cyclic group has a unique subgroup of every order dividing the order of the group. \item State the structure theorem for finitely generated abelian groups. Characterize up to isomorphism all abelian groups of order 72 both in terms of elementary divisors and invariant factors. \item Show that every finite abelian group is the direct product of its Sylow $p$-subgroups. \item Show that every group of order 56 is the semidirect product of its Sylow $p$-subgroups. \item Let $G$ be a finite group and $H$ a normal subgroup. If $\gcd(|H|,[G:H]) = 1$, show that $H$ is the unique subgroup of $G$ with order $|H|$. \item State the Jordan-H\"older theorem. Give an example of two nonisomorphic groups with the same composition factors. \item Why is the classification of all the finite simple groups such a big deal? That is, how does it fit into the grand scheme of classifying all the finite groups? Hint: Suppose that $G$ is a finite group with composition series $G \triangleright K \triangleright \{1\}$. This gives us a natural short exact sequence: $1 \longrightarrow K \longrightarrow G \longrightarrow G/K \longrightarrow 1$ in which we have information about $K$ and $G/K$. Now go read about the ``extension problem'', and think about the general case. \end{enumerate} %%%%%%%%%%%%% \section{Ring Theory} \begin{enumerate} \item Define the terms integral domain, irreducible element, and prime element. \begin{enumerate} \item Are there rings with no irreducible elements? \item Are there commutative rings in which irreducible elements are not prime? \item Prove that in an integral domain, prime elements are irreducible. \end{enumerate} \item Give examples of a noncommutative ring with zero divisors, a noncommutative division ring, and integral domain, a UFD, a PID, a Euclidean domain and examples which show that ID $\not\Rightarrow$ UFD $\not\Rightarrow$ PID $\not\Rightarrow$ ED. Be sure to justify that your examples have or do not have the requisite properties. \item Define what is meant by a Noetherian ring. Give three equivalent conditions and demonstrate their equivalence. \item Why isn't every integral domain a UFD? That is, what goes wrong when one tries to factor nonzero nonunits. What conditions does one need to impose? Hint: there are separate conditions which guarantee the existence of a factorization and uniqueness of a factorization. \item What is the characteristic of a commutative ring with identity? What can one say about the characteristic of an integral domain? \item Prove that a Euclidean integral domain is a PID. \item Show that a PID is a UFD. \item Let $A$ be a UFD, and $X$ an indeterminate. Show that any irreducible in $A$ remains irreducible in $A[X]$. \item Show that $k[x_1, x_2, \ldots]$ is a non-Noetherian UFD. Be careful here. The Noetherian issue is trivial, but the UFD part is not. For example what are the irreducibles? Does a polynomial which is irreducible in $k[x_1,\ldots,x_n]$ remain irreducible in $k[x_1, x_2, \ldots]$? \item Prove that over a field $K$, a polynomial of degree $n$ has at most $n$ roots in any splitting field. Does this remain true if the field $K$ is replaced by a division ring, like Hamilton's quaternions? Why or why not? \item Let $A$ be a UFD with quotient field $K$, and let $f\in A[X]$. Show that $f$ is irreducible over $A$ iff $f$ is primitive and irreducible over $K[X]$. \item Let $A$ be a UFD with quotient field $K$, and let $L$ be a field. Let $f \in A[X]$, $\deg(f) = r \ge 1$, and let $\sigma:A \to L$ be a ring homomorphism. Show that if $\deg(f^\sigma) = \deg(f)$ and $f^\sigma$ is irreducible in $L[X]$, then $f$ is irreducible in $K[X]$. In particular, if $f$ is primitive, then $f$ is irreducible in $A[X]$. \item Use the previous question to show that $f(x) = x^3 + 3x^2 + 5x + 2$ is irreducible over $\Z$. (Let $\sigma:\Z \to \Z/3\Z$ be the natural map) \item State and prove Eisenstein's criterion for irreducibility. \item State and prove the Chinese Remainder theorem in the context of a commutative ring with identity. Give an interpretation in terms of $\Z/N\Z$ and of $\left(\Z/N\Z\right)^\times$. (Recall that if $I$, $J$ are ideals in a commutative ring that $IJ \subset I \cap J$ and $(I\cap J) (I + J) \subset IJ$. Now use comaximality) \item Let $A$ be a ring, and $X$ an indeterminate. For each property $P$ listed below, consider the question: ``If $A$ has $P$, does $A[X]$?'' If so, give a proof; if not a counterexample. \begin{enumerate} \item integral domain \item PID \item UFD \item Noetherian ring \end{enumerate} \item Define the notion of prime and maximal ideals. Give characterizations of each concept in terms of quotient rings. Give an example to show that not all prime ideals are maximal. \begin{enumerate} \item Is $(x)$ prime/maximal is $\Z[x],\ \Q[x]$? \item If $k$ is a field, is $(x)$ prime or maximal in $k[x,y]$? \item Characterize the maximal ideals of $\Q[x]$, or $\C[x]$. \item Is $(x-3)$ prime/maximal in $k[x,y]$? \end{enumerate} \item Define what is meant by the localization of a ring with respect to a multiplicative set. What does localizing at a prime mean? Describe the localization of the ring $\Z[x]$ at the prime ideal $(x)$. Is $(x)$ maximal in $\Z[x]_{(x)}$? If so, describe the field to which its quotient is isomorphic. How does $\Z[x]_{(x)}$ compare to $\Q[x]_{(x)}$? \item Show that every ideal of the localized ring $S^{-1}A$ is of the form $S^{-1}I$ where $I$ is an ideal of $A$. Use this to show that if $A$ is a PID, then so is $S^{-1}A$. (Hint: if $[a,s]$ is in an ideal of $S^{-1}A$, then so is $[a,1]$ and hence so is $[a,t]$ for every $t \in S$.) \item If $A$ is a commutative ring with identity and the polynomial ring $A[x]$ is Noetherian, does it follow that $A$ is Noetherian? This is the converse of a well-known theorem. \end{enumerate} %%%%%%%%%%%%%%%%% \section{Modules and Linear Algebra} \begin{enumerate} \item Discuss the notion of a minimal polynomial of a linear transformation. Does it have to be irreducible? \item Let $V$ be a finite-dimensional vector space over the field $k$, and let $T \in {\rm End}_k(V)$. Show how to use $T$ to make $V$ into a finitely generated torsion $k[x]$-module. \item State the basic decomposition theorem for finitely generated modules over PIDs and apply it to the situation in the previous question. \item Explain the origins of the rational canonical form of a matrix. \item Describe what is meant by the Jordan form of a matrix. \item Find all rational and Jordan canonical forms of a matrix in $M_5(\C)$ having minimal polynomial $x^2(x-1)$. Be sure to give the corresponding invariants and the characteristic polynomial. \item Show that any linear operator $T$ on a finite dimensional vector space (over a field of characteristic not equal to 2) which satisfies $T^2 = I$ is diagonalizable. \item State and prove the rank-nullity theorem for finite dimensional vector spaces. \item For a vector space over a field, we know that any linearly independent set can be extended to a basis, and that any spanning set can be reduced to a basis. Let $M$ be a free module of finite rank over a PID $R$. Are the corresponding statements for $M$ valid? Proof or counter example. \item Let $M$ be a free module of finite rank over a PID $R$ and $N$ a submodule. What does the elementary divisor theorem say about bases of $N$ versus $M$? \item Let $M$ and $N$ be modules over a commutative ring $R$. \begin{enumerate} \item Characterize the tensor product $M\otimes_R N$ as a universal object in some category. \item Give a construction for the tensor product $M\otimes_R N$. \item Show that $\Z/m\Z \otimes_\Z \Z/n\Z = 0$ if $\gcd(m,n) = 1$. \item Characterize $\Z/m\Z \otimes_\Z \Z/n\Z$ in general. \item If $G$ is a finite abelian group, and $G \otimes_\Z \Z/p\Z = 0$ for all primes $p$, show that $G= 0$. Does the result remain true if $G$ is infinite. \end{enumerate} \item Let $A$ be a commutative ring with identity, and suppose that $0 \rightarrow M \rightarrow N \rightarrow F \rightarrow 0$ is a short exact sequence of $A$-modules. Show that if $F$ is free, the sequence splits. Find examples of $A, M,N,F$ so the short exact sequence doesn't split. What are the minimal conditions to place of $F$ so that the sequence splits? What can be said about the relationship between $M,N$ and $F$. \item Give an example of a free-module and a submodule which is not free. What conditions guarantee that a submodule of a free module be free? \end{enumerate} %%%%%%%%%%%%%%%% \section{Field Theory} \begin{enumerate} \item Let $E/F$ be an extension of fields, and let $\alpha \in E$. Define what is meant by the phrase $\alpha$ is algebraic over $F$. What is meant by saying that $E/F$ is an algebraic extension? \item What is meant by the minimal polynomial of an element algebraic over $F$? Why is $F[\alpha] \cong F(\alpha)$ when $\alpha$ is algebraic over $F$? \item If $\alpha$ is algebraic over $F$, what is the degree of $F(\alpha)/F$? Exhibit a basis and prove that it is a basis. \item Show that an extension of fields $E/F$ is finite if and only if it is algebraic and finitely generated. \item Consider a tower of fields $K \subset F \subset E$. Show that $E/K$ is finite (resp. algebraic) if and only if $E/F$ and $F/K$ are finite (resp. algebraic). Consider the situation for normal extensions. \item Give three equivalent conditions for an extension to be normal (don't restrict to finite extensions), and demonstrate their equivalence. \begin{enumerate} \item Every embedding $\sigma:E/F \to \bar F$ is an automorphism of $E$. \item $E$ is the splitting field of a family of polynomials in $F[X]$. \item Every irreducible polynomial in $F[X]$ with one root in $E$ splits in $E$. \end{enumerate} \item Show that finite fields are perfect. \item Let $F$ be a field, and $f \in F[x]$ be irreducible. Show that all the roots of $f$ occur with the same multiplicity. If $F$ is perfect, show that all roots occur with multiplicity one. \item Prove that the Galois group of a finite extension of finite fields is cyclic. \item Compute the Galois group of the splitting field of $x^8 - 1$ over $\Z_3$. Is $x^8 - 1$ separable over $\Z_3$? Do you really need to check? Hint: $x^9 - x = x ( x^8 - 1)$. \item Describe the Galois group of $\Q(\zeta_n)/\Q$ and discuss the Galois correspondence. Here $\zeta_n$ is a primitive $n$th root of unity. \item Describe all the fields which lie between $\Q$ and ${\Q}(\zeta_{12})$. \item Define what is meant by a purely inseparable extension of fields. If $k(\alpha)/k$ is purely inseparable, verify that any embedding $\sigma:k(\alpha) \to \bar k$ (over $k$) is the identity map. \item Let $f \in F[x]$ be a nonconstant polynomial. Show that there is a field extension of $F$ in which $f$ has a root. \item Describe the construction of an algebraic closure of a field. \item Show that any finite subgroup of the multiplicative group of a field is cyclic. \item Let $E/F$ be an extension of fields and let $\alpha \in E$ be algebraic over $F$. If $[F(\alpha):F]$ is odd, show that $F(\alpha) = F(\alpha^2)$. \item Suppose that $\xi$ is transcendental over a field $F$. Let $E$ be a subfield of $F(\xi)$ which is not equal to $F$. \begin{enumerate} \item Show that $\xi$ is algebraic over $E$. \item Show that $F(\xi)$ is a finite extension of $E$. \item Is $E$ algebraic over $F$? \end{enumerate} \item Compute the Galois group of $x^6 + 27$ over $\Q$. \item Compute the Galois group of $x^8 - 16$ over $\Q$. \end{enumerate} \end{document}