## A lot of algebra prelims

Contents\
1 2018 2\
2 2017 3\
3 2016 4\
4 2015 5\
5 2014 6\
6 2013 8\
7 2012 9\
8 2011 10\
9 2010 11\
10 2009 13\
11 2008 15\
12 2007 16\
13 2006 17\
14 2005 18

## 

## 1 2018

Problem 1 (2018, 1). Show that no finite group is the union of conjugates of a proper subgroup.

Problem 2 (2018, 2). Classify all groups of order 18 up to isomorphism.

Problem 3 (2018, 3). Let $\alpha , \beta$ denote the unique positive real $5 ^ { \mathrm { t h } }$ root of 7 and $4 ^ { \mathrm { t h } }$ root of 5, respectively.\
Determine the degree of $\mathbb { Q } ( \alpha , \beta )$ over Q.

Problem 4 (2018, 4). Show that the field extension $\mathbb { Q } \subseteq \mathbb { Q } \left( { \sqrt { 2 + { \sqrt { 2 } } } } \right)$ is Galois and determine its Galois group.

Problem 5 (2018, 5). Let M be a square matrix over a field K. Use a suitable canonical form to show that M is similar to its transpose $M ^ { T }$

Problem 6 (2018, 7). Let G be a finite group and $\pi , \pi ^ { \prime }$ be two irreducible representations of G. Prove or disprove the following assertion: π and $\pi ^ { \prime }$ are equivalent if and only if det $\pi ( g ) = \operatorname* { d e t } \pi ^ { \prime } ( g )$ for all $g \in G$

## 2 2017

Problem 1 (2017, 1). Let R be a Noetherian ring.
Prove that $R [ x ]$ and $R [ [ x ] ]$ are both Noetherian.
(The first part of the question is asking you to prove the Hilbert Basis Theorem, not to use it!)

Problem 2 (2017, 2). Classify (with proof) all fields with finitely many elements.

Problem 3 (2017, 3). Suppose A is a commutative ring and M is a finitely presented module.
Given any surjection φ : An → M from a finite free A-module, show that ker φ is finitely generated.

Problem 4 (2017, 4.). Classify all groups of order 57.

Problem 5 (2017, 7). Show that a finite simple group cannot have a 2-dimensional irreducible representation over C. (Hint: the determinant might prove useful.)

## 3 2016

Problem 1 (2016, 1.). Let G be a finite simple group.
Assume that every proper subgroup of G is abelian.\
Prove that then G is cyclic of prime order.

Problem 2 (2016, 2). Let $a \in \mathbb { N } , a > 0 .$ . Compute the Galois group of the splitting field of the polynomial $x ^ { 5 } - 5 a ^ { 4 } x + a { \mathrm { ~ o v e r ~ Q } }$

Problem 3 (2016, 4.). Recall that an inner automorphism of a group is an automorphism given by conjugation by an element of the group.
An outer automorphism is an automorphism that is not inner.

(a) Prove that $S _ { 5 }$ has a subgroup of order 20.

(b) Use the subgroup from (a) to construct a degree 6 permutation representation of $S _ { 5 } \ ( \mathrm { i . e . }$ , an embedding $S _ { 5 } \hookrightarrow S _ { 6 }$ as a transitive permutation group on 6 letters).

(c) Conclude that $S _ { 6 }$ has an outer automorphism.

Problem 4 (2016, 5.). Let A be a commutative ring and M a finitely generated A-module.
Define

$$
\operatorname { A n n } ( M ) = \{ a \in A : a m = 0 { \mathrm { ~ f o r ~ a l l ~ } } m \in M \} .
$$

Show that for a prime ideal ${ \mathfrak { p } } \subset A$ , the following are equivalent:

(a) $\mathsf { A n n } ( M ) \not \subset \mathfrak { p }$

(b) The localization of M at the prime ideal p is 0.

(c) $M \otimes _ { A } k ( { \mathfrak { p } } ) = 0$ , where $k ( { \mathfrak { p } } ) = A _ { \mathfrak { p } } / { \mathfrak { p } } A _ { \mathfrak { p } }$ is the residue field of A at p.

Problem 5 (2016, 6.). Let $A = \mathbb { C } [ x , y ] / ( y ^ { 2 } - ( x - 1 ) ^ { 3 } - ( x - 1 ) ^ { 2 } )$

(a) Show that A is an integral domain and sketch the R-points of SpecA.

(b) Find the integral closure of A. Recall that for an integral domain A with fraction field K, the integral closure of A in K is the set of all elements of K integral over A.

Problem 6. Let $R = k [ x , y ]$ where k is a field, and let $\boldsymbol { I } = ( x , y ) \boldsymbol { R }$

1. Show that

$$
0 \xrightarrow [ ] { } R \xrightarrow [ ] { \phi } R \oplus R \xrightarrow [ ] { \psi } R \longrightarrow k \xrightarrow [ ] { } 0
$$

where $\phi ( a ) = ( - y a , x a ) , \psi ( ( a , b ) ) = x a + y b$ for $a , b \in R ,$ , is a projective resolution of the R-module $k \simeq R / I .$

2. Show that I is not a flat R-module by computing $\mathrm { T o r } _ { i } ^ { R } ( I , k )$

## 4 2015

Problem 1 (2015, 1). (a) Find an irreducible polynomial of degree 5 over the field $\mathbb { Z } / 2$ of two elements and use it to construct a field of order 32 as a quotient of the polynomial ring $\mathbb { Z } / 2 [ x ]$

(b) Using the polynomial found in part (a), find a $5 \times 5$ matrix M over $\mathbb { Z } / 2$ of order 31, so that $M ^ { 3 1 } = I$ but $M \neq I .$

Problem 2 (2015, 2). Find the minimal polynomial of ${ \sqrt { 2 } } + { \sqrt { 3 } }$ over Q. Justify your answer.

Problem 3 (2015, 3). (a) Let R be a commutative ring with no nonzero nilpotent elements.
Show that the only units in the polynomial ring $R [ x ]$ are the units of $R ,$ regarded as constant polynomials.

(b) Find all units in the polynomial ring $\mathbb { Z } _ { 4 } [ x ]$

Problem 4 (2015, 4.). Let $p , q$ be two distinct primes.
Prove that there is at most one non-abelian group of order $p q$ and describe the pairs $\left( p , q \right)$ such that there is no non-abelian group of order $p q$

Problem 5 (2015, 5). (a) Let L be a Galois extension of a field K of degree 4. What is the minimum number of subfields there could be strictly between K and L? What is the maximum number of such subfields?
Give examples where these bounds are attained.

(b) How do these numbers change if we assume only that L is separable (but not necessarily Galois) over K?

Problem 6 (2015, 6). Let R be a commutative algebra over C. A derivation of R is a C-linear map D : $R \to R$ such that (i) $D ( 1 ) = 0$ and (ii) $D ( a b ) = D ( a ) b \dot { + } a D ( b )$ for all $a , b \in R$

(a) Describe all derivations of the polynomial ring $\mathbb { C } [ x ]$

(b) Let A be the subring (or C-subalgebra) of ${ \mathrm { E n d } } _ { \mathbb { C } } ( \mathbb { C } [ x ] )$ generated by all derivations of $\mathbb { C } [ x ]$ and the left multiplications by x. Prove that $\mathbb { C } [ x ]$ is a simple left A-module.
Note that the inclusion $A $ ${ \mathrm { E n d } } _ { \mathbb { C } } ( \mathbb { C } [ { \bar { x } } ] )$ defines a natural left A-module structure on C[x].

Problem 7 (2015, 7). Let G be a non-abelian group of order $p ^ { 3 }$ with p a prime.

(a) Determine the order of the center $Z$ of G.

(b) Determine the number of inequivalent complex 1-dimensional representations of G.

(c) Compute the dimensions of all the inequivalent irreducible representations of G and verify that the number of such representations equals the number of conjugacy classes of G.

## 5 2014

Problem 1 (2014, 1.). (a) Let G be a group (not necessarily finite) that contains a subgroup of index n. Show that G contains a normal subgroup N such that $n \leq [ G : N ] \leq n !$

(b) Use part (a) to show that there is no simple group of order 36.

Problem 2 (2014, 2). Let p be a prime, let $\mathbb { F } _ { p }$ be the p-element field, and let $K = \mathbb { F } _ { p } ( t )$ be the field of rational functions in t with coefficients in $\mathbb { F } _ { p }$ . Consider the polynomial $f ( x ) = x ^ { p } - t \stackrel { . } { \in } K [ x ]$ .

(a) Show that f does not have a root in K.

(b) Let E be the splitting field of f over K. Find the factorization of f over E.

(c) Conclude that f is irreducible over K.

Problem 3 (2014, 3). Recall that a ring A is called graded if it admits a direct sum decomposition $A =$ $\textstyle \bigoplus _ { n = 0 } ^ { \infty } A _ { n }$ as abelian groups, with the property that $A _ { i } A _ { j } \subseteq A _ { i + j }$ for all $i , j \geq 0$ . Prove that a graded commutative ring $A = \oplus _ { n = 0 } ^ { \infty } A _ { n }$ is Noetherian if and only if $A _ { 0 }$ is Noetherian and A is finitely generated as an algebra over $A _ { 0 }$

Problem 4 (2014, 4). Let R be a ring with the property that $a ^ { 2 } = a$ for all $a \in R$

(a) Compute the Jacobson radical of R.

(b) What is the characteristic of R?

(c) Prove that R is commutative.

(d) Prove that if R is finite, then R is isomorphic (as a ring) to $( \mathbb { Z } / 2 \mathbb { Z } ) ^ { d }$ for some d.

Problem 5 (2014, 6). Let $\overline { { \mathbb { F } _ { p } } }$ denote the algebraic closure of $\mathbb { F } _ { p }$ . Show that the Galois group $\operatorname { G a l } ( { \overline { { \mathbb { F } _ { p } } } } / \mathbb { F } _ { p } )$ has no non-trivial finite subgroups.

Problem 6 (2014, 7). Let $C _ { p }$ denote the cyclic group of order $p .$

(a) Show that $C _ { p }$ has two irreducible representations over Q (up to isomorphism), one of dimension 1 and one of dimension $p - 1$

(b) Let G be a finite group, and let $\rho : G \to { \mathrm { G L } } _ { n } ( \mathbb { Q } )$ be a representation of G over Q. Let $\rho _ { \mathbb { C } } : G \to \mathbf { G } \mathrm { L } _ { n } ( \mathbb { C } )$ denote $\rho$ followed by the inclusion ${ \mathrm { G L } } _ { n } ( \mathbb { Q } ) \to { \mathrm { G L } } _ { n } ( \mathbb { C } )$ . Thus $\rho _ { \mathbb { C } }$ is a representation of G over C, called the complexification of $\rho .$ . We say that an irreducible representation $\rho$ of G is absolutely irreducible if its complexification remains irreducible over C.

Now suppose G is abelian and that every representation of G over Q is absolutely irreducible.
Show that $G \cong { \bar { ( } } C _ { 2 } ) ^ { k }$ for some $k ( \mathrm { i . e . , }$ , is a product of cyclic groups of order 2).

Problem 7 (2014, 8). Let G be a finite group and $\mathbb { Z } [ G ]$ the internal group algebra.
Let $\mathcal { Z }$ be the center of Z[G]. For each conjugacy class $C \subseteq G$ , let $\begin{array} { r } { P _ { C } = \sum _ { g \in C } g } \end{array}$

(a) Show that the elements $P _ { C }$ form a Z-basis for Z. Hence ${ \mathcal Z } \cong \mathbb { Z } ^ { d }$ as an abelian group, where d is the number of conjugacy classes in G.

(b) Show that if a ring R is isomorphic to $\mathbb { Z } ^ { d }$ as an abelian group, then every element in R satisfies a monic integral polynomial.
(Hint: Let $\{ v _ { 1 } , \ldots , v _ { d } \}$ be a basis of R and for a fixed non-zero $r \in R$ , write $\begin{array} { r } { r v _ { i } = \sum _ { j } a _ { i j } v _ { j } } \end{array}$ . Use the Hamilton-Cayley theorem.)

(c) Let $\pi : G \to \mathbf { G } \mathbf { L } ( V )$ be an irreducible representation of G (over C). Show that $\pi ( P _ { C } )$ acts on V as multiplication by the scalar

$$
{ \frac { | C | \chi _ { \pi } ( C ) } { \dim V } } ,
$$

where $\chi _ { \pi } ( C )$ is the value of the character $\chi _ { \pi }$ on any element of C.

(d) Conclude that $| C | \chi _ { \pi } ( C ) /$ dim V is an algebraic integer.

## 6 2013

Problem 1 (2013, 3.). (a) Suppose that G is a finitely generated group.
Let n be a positive integer.
Prove that G has only finitely many subgroups of index n

(b) Let p be a prime number.
If G is any finitely-generated abelian group, let $t _ { p } ( G )$ denote the number of subgroups of G of index $p .$ Determine the possible values of $t _ { p } ( G )$ as G varies over all finitelygenerated abelian groups.

Problem 2 (2013, 4.). Suppose that G is a finite group of order 2013. Prove that G has a normal subgroup N of index 3 and that N is a cyclic group.
Furthermore, prove that the center of G has order divisible by 11. (You will need the factorization $2 0 1 3 = 3 \cdot 1 1 \cdot 6 1 . )$

Problem 3 (2013, 6). This question concerns an extension K of Q such that $\left[ K : \mathbb { Q } \right] = 8$ . Assume that $K / \mathbb { Q }$ is Galois and let $G = { \mathrm { G a l } } ( K / \mathbb { Q } )$ . Furthermore, assume that G is non-abelian.

(a) Prove that K has a unique subfield F such that $F / \mathbb { Q }$ is Galois and $[ F : \mathbb { Q } ] = 4 .$

(b) Prove that F has the form $F = \mathbb { Q } ( { \sqrt { d _ { 1 } } } , { \sqrt { d _ { 2 } } } )$ where $d _ { 1 } , d _ { 2 }$ are non-zero integers.

(c) Suppose that G is the quaternionic group.
Prove that $d _ { 1 }$ and $d _ { 2 }$ are positive integers.

Problem 4 (2013, 8). This question concerns the polynomial ring $R = \mathbb { Z } [ x , y ]$ and the ideal $I = ( 5 , x ^ { 2 } + 2 )$ in R.

(a) Prove that I is a prime ideal of R and that R/I is a PID.

(b) Give an explicit example of a maximal ideal of R which contains I. (Give a set of generators for such an ideal.)

(c) Show that there are infinitely many distinct maximal ideals in R which contain I.

## 7 2012

Problem 1 (2012, 1.). Classify all groups of order 2012 up to isomorphism.
(Hint: 503 is prime).

Problem 2 (2012, 2.). For any positive integer n, let $G _ { n }$ be the group generated by a and b subject to the following three relations:

$$
a ^ { 2 } = 1 , \quad b ^ { 2 } = 1 , \quad { \mathrm { a n d } } \quad ( a b ) ^ { n } = 1 .
$$

(a) Find the order of the group $G _ { n }$

(We don’t know how to do the rest of the problem)

Problem 3 (2012, 6). Determine the Galois groups of the following polynomials over Q.

(a) $f ( x ) = x ^ { 4 } + 4 x ^ { 2 } + 1$

(b) $f ( x ) = x ^ { 4 } + 4 x ^ { 2 } - 5 .$

Problem 4 (2012, 3). Let R be a (commutative) principal ideal domain, let M and N be finitely generated free R-modules, and let $\varphi : M \to N$ be an R-module homomorphism.

(a) Let K be the kernel of ϕ. Prove that K is a direct summand of M.

(b) Let C be the image of ϕ. Show by example (specifying R, M, N, and ϕ) that C need not be a direct summand of N.

## 8 2011

Problem 1 (2011, 2.). In this problem, as you apply Sylow’s Theorem, state precisely which portions you are using.

(a) Prove that there is no simple group of order 30.

(b) Suppose that G is a simple group of order 60. Determine the number of p-Sylow subgroups of G for each prime p dividing 60, then prove that G is isomorphic to the alternating group $A _ { 5 }$

Note: in the second part, you needn’t show that $A _ { 5 }$ is simple.
You need only show that if there is a simple group of order 60, then it must be isomorphic to $A _ { 5 }$

Problem 2 (2011, 3). Describe the Galois group and the intermediate fields of the cyclotomic extension $\mathbb { Q } ( \zeta _ { 1 2 } ) / \mathbb { Q }$

Problem 3 (2011, 4). Let

$$
R = \mathbb { Z } [ x ] / ( x ^ { 2 } + x + 1 ) .
$$

(a) Answer the following questions with suitable justification.

(i) Is R a Noetherian ring?

(ii) Is R an Artinian ring?

(b) Prove that R is an integrally closed domain.

Problem 4 (2011, 5). Let R be a commutative ring.
Recall that an element r of R is nilpotent if $r ^ { n } = 0$ for some positive integer n and that the nilradical of R is the set $N ( R )$ of nilpotent elements.

(a) Prove that

$$
N ( R ) = \cap _ { P \mathrm { { p r i m e } } } P .
$$

(Hint: given a non-nilpotent element r of R, you may wish to construct a prime ideal that does not contain r or its powers.)

(b) Given a positive integer m, determine the nilradical of $\mathbb { Z } / ( m )$

(c) Determine the nilradical of $\mathbb { C } [ x , y ] / ( y ^ { 2 } - x ^ { 3 } )$

(d) Let $p ( x , y )$ be a polynomial in $\mathbb { C } [ x , y ]$ such that for any complex number $a , p ( a , a ^ { 3 / 2 } ) = 0$ . Prove that $p ( x , y )$ is divisible by $y ^ { 2 } - x ^ { 3 }$

Problem 5 (2011, 6). Given a finite group $G ,$ recall that its regular representation is the representation on the complex group algebra C[G] induced by left multiplication of G on itself and its adjoint representation is the representation on the complex group algebra C[G] induced by conjugation of G on itself.

(a) Let $G = { \mathrm { G L } } _ { 2 } ( \mathbb { F } _ { 2 } )$ . Describe the number and dimensions of the irreducible representations of G. Then describe the decomposition of its regular representation as a direct sum of irreducible representations.

(b) Let G be a group of order 12. Show that its adjoint representation is reducible; that is, there is an H-invariant subspace of C[H] besides 0 and $\mathbb { C } [ H ]$

Problem 6 (2011, 8). Let R be a commutative integral domain.
Show that the following are equivalent:

(a) R is a field;

(b) R is a semi-simple ring;

(c) Any R-module is projective.

## 9 2010

Problem 1 (2010, 1.). Let $p$ be a positive prime number, $\mathbb { F } _ { p }$ the field with p elements, and let $G = \mathrm { G L } _ { 2 } ( \mathbb { F } _ { p } )$

(a) Compute the order of $G , | G |$

(b) Write down an explicit isomorphism from $\mathbb { Z } / p \mathbb { Z }$ to

$$
U = { \Bigg \{ } { \binom { 1 } { 0 } } \ { \stackrel { a } { 1 } } { \Bigg | } a \in \mathbb { F } _ { p } { \Bigg \} } .
$$

(c) How many subgroups of order p does G have?
Hint: compute $g u g ^ { - 1 }$ for $g \in G$ and $u \in U ;$ use this to find the size of the normalizer of U in G.

Problem 2 (2010, 2). (a) Give definitions of the following terms: (i) a finite length (left) module, (ii) a composition series for a module, and (iii) the length of a module,

(b) Let l(M) denote the length of a module M. Prove that if

$$
0 \to M _ { 1 } \to M _ { 2 } \to \cdots \to M _ { n } \to 0
$$

is an exact sequence of modules of finite length, then

$$
\sum _ { i = 1 } ^ { n } ( - 1 ) ^ { k } l ( M _ { i } ) = 0 .
$$

Problem 3 (2010, 3). Let F be a field of characteristic $p ,$ and G a group of order $p ^ { n }$ . Let $R = \mathbb { F } [ G ]$ be the group ring (group algebra) of G over F, and let $\textstyle u : = \sum _ { x \in G } x$ (so u is an element of R).

(a) Prove that u lies in the center of R.

(b) Verify that Ru is a 2-sided ideal of R.

(c) Show there exists a positive integer k such that $u ^ { k } = 0$ . Conclude that for such a k, $( R u ) ^ { k } = 0 .$

(d) Show that R is not a semi-simple ring.
(Warning: Please use the definition of a semi-simple ring: do not use the result that a finite length ring fails to be semisimple if and only if it has a non-zero nilpotent ideal.)

Problem 4 (2010, 4). Let $f ( x ) = a _ { n } x ^ { n } + a _ { n - 1 } x ^ { n - 1 } + \cdot \cdot \cdot + a _ { 0 } \in \mathbb { Z } [ x ]$ (where $a _ { n } \neq 0 )$ and let $R = \mathbb { Z } [ x ] / ( f )$ Prove that R is a finitely generated module over Z if and only if $a _ { n } = \pm 1$ .

Problem 5 (2010, 5). Consider the ring

$$
S = C [ 0 , 1 ] = \{ f : [ 0 , 1 ] \to \mathbb { R } : f \ \mathrm { i s ~ c o n t i n u o u s } \}
$$

with the usual operations of addition and multiplication of functions.

(a) What are the invertible elements of S?

(b) For $a \in [ 0 , 1 ] .$ , define $I _ { a } = \{ f \in S : f ( a ) = 0 \}$ . Show that $I _ { a }$ is a maximal ideal of S.

(c) Show that the elements of any proper ideal of S have a common zero, i.e., if I is a proper ideal of $S ,$ then there exists $a \in [ 0 , 1 ]$ such that $f ( a ) = 0$ for all $f \in I .$ Conclude that every maximal ideal of S is of the form $I _ { a }$ for some $a \in [ 0 , 1 ]$ . Hint: as [0, 1] is compact, every open cover of $[ 0 , 1 ]$ contains a finite subcover.

Problem 6 (2010, 7). Let F be a field of characteristic zero, and let K be an algebraic extension of F that possesses the following property: every polynomial $f \in F [ x ]$ has a root in K. Show that K is algebraically closed.

Hint: if $K ( \theta ) / K$ is algebraic, consider $F ( \theta ) / F$ and its normal closure; primitive elements might be of help.

Problem 7 (2010, 8). Let G be the unique non-abelian group of order 21.

(a) Describe all 1-dimensional complex representations of G.

(b) How many (non-isomorphic) irreducible complex representations does G have and what are their dimensions?

(c) Determine the character table of G.

## 10 2009

Problem 1 (2009, 1.). (a) Classify all groups of order $2 0 0 9 = 7 ^ { 2 } \times 4 1$

(b) Suppose that G is a group of order 2009. How many intermediate groups are there—that is, how many groups H are there with $1 \subsetneq H \subsetneq G .$ , where both inclusions are proper?
(There may be several cases to consider.)

Problem 2 (2009, 2). Let K be a field.
A discrete valuation on K is a function $\nu : K \setminus \{ 0 \} \to \mathbb { Z }$ such that

(i) $\nu ( a b ) = \nu ( a ) + \nu ( b )$

(ii) ν is surjective

(iii) $\nu ( a + b ) \geq \operatorname* { m i n } \{ ( \nu ( a ) , \nu ( b ) \}$ for $a , b \in K \backslash \{ 0 \}$ with $a + b \neq 0$

Let $R : = \left\{ x \in K \setminus \{ 0 \} : \nu ( x ) \geq 0 \right\} \cup \{ 0 \}$ . Then R is called the valuation ring of ν. Prove the following:

(a) R is a subring of K containing the 1 in K.

(b) for all $x \in K \backslash \{ 0 \}$ , either $x \mathrm { o r } x ^ { - 1 }$ is in R.

(c) x is a unit of R if and only if $\nu ( x ) = 0$

(d) Let p be a prime number, $K = \mathbb { Q }$ , and $\nu _ { p } : \mathbb { Q } \backslash \{ 0 \}  \mathbb { Z }$ be the function defined by $\nu _ { p } ( { \frac { a } { b } } ) = n$ where $\begin{array} { r } { \frac { a } { b } = p ^ { n } \frac { c } { d } } \end{array}$ and p does not divide c and d. Prove that the corresponding valuation ring R is the ring of all rational numbers whose denominators are relatively prime to p.

Problem 3 (2009, 3). Let F be a field of characteristic not equal to 2.

(a) Prove that any extension K of F of degree 2 is of the form $F ( \sqrt { D } )$ where $D \in F$ is not a square in F and, conversely, that each such extension has degree 2 over F.

(b) Let $D _ { 1 } , D _ { 2 } \in F$ neither of which is a square in F. Prove that $[ F ( \sqrt { D _ { 1 } } , \sqrt { D _ { 2 } } ) : F ] = 4 \mathrm { i f } D _ { 1 } D _ { 2 }$ is not a square in F and is of degree 2 otherwise.

Problem 4 (2009, 4). Let F be a field and $p ( x ) \in F [ x ]$ an irreducible polynomial.

(a) Prove that there exists a field extension K of F in which $p ( x )$ has a root.

(b) Determine the dimension of K as a vector space over F and exhibit a vector space basis for K.

(c) If $\theta \in K$ denotes a root of p(x), express $\theta ^ { - 1 }$ in terms of the basis found in part (b).

(d) Suppose $p ( x ) = x ^ { 3 } + 9 x + 6$ . Show $p ( x )$ is irreducible over Q. If θ is a root of $p ( x )$ , compute the inverse of (1 + θ) in Q(θ).

Problem 5 (2009, 6). Fix a ring R, an R-module M, and an R-module homomorphism $f : M \to M .$

(a) If M satisfies the descending chain condition on submodules, show that if f is injective, then f is surjective.
(Hint: note that if f is injective, so are $f \circ f , f \circ f \circ f ,$ etc.)

(b) Give an example of a ring R, an R-module M, and an injective R-module homomorphism $f : M \to M$ which is not surjective.

(c) If M satisfies the ascending chain condition on submodules, show that if f is surjective, then f is injective.

(d) Give an exampe of a ring R, and R-module M, and a surjective R-module homomorphism $f : M \to M$ which is not injective.

Problem 6 (2009, 7). Let G be a finite group, k an algebraically closed field, and V an irreducible k-linear representation of G.

(a) Show that ${ \mathrm { H o m } } _ { k G } ( V , V )$ is a division algebra with k in its center.

(b) Show that V is finite-dimensional over $k ,$ and conclude that ${ \mathrm { H o m } } _ { k G } ( V , V )$ is also finite dimensional.

(c) Show the inclusion $k \hookrightarrow \operatorname { H o m } _ { k G } ( V , V )$ found in (a) is an isomorphism.
(For $f \in { \mathrm { H o m } } _ { k G } ( V , V )$ , view $f$ as a linear transformation and consider $f - \alpha I ,$ , where α is an eigenvalue of f ).

## 11 2008

Problem 1 (2008, 1). Let $f ( x )$ be an irreducible polynomial of degree 5 over the field Q of rational numbers with exactly 3 real roots.

(a) Show that $f ( x )$ is not solvable by radicals.

(b) Let E be the splitting field of f over Q. Construct a Galois extension K of degree 2 over Q lying in E such that no field F strictly between K and E is Galois over Q.

Problem 2 (2008, 2). Let F be a finite field.
Show for any positive integer n that there are irreducible polynomials of degree n in $F [ x ]$

Problem 3 (2008, 3.). Show that the order of the group ${ \mathrm { G L } } _ { n } ( \mathbb { F } _ { q } )$ of invertible $n \times n$ matrices over the field $\mathbb { F } _ { q }$ of q elements is given by $( q ^ { n } - 1 ) ( q ^ { n } - q ) \dots ( \bar { q } ^ { n } - \bar { q } ^ { n - 1 } )$

Problem 4 (2008, 5). (a) Let R be a commutative principal ideal domain.
Show that any R-module M generated by two elements takes the form $R / ( a ) \oplus R / ( b )$ for some $a , b \in R$ . What more can you say about a and b?

(b) Give a necessary and sufficient condition for two direct sums as in part (a) to be isomorphic as Rmodules.

Problem 5 (2008, 6.). Let G be the subgroup of ${ \mathrm { G L } } _ { 3 } ( \mathbb { C } )$ generated by the three matrices

$$
A = { \left( \begin{array} { l l l } { 0 } & { 0 } & { 1 } \\ { 0 } & { 1 } & { 0 } \\ { 1 } & { 0 } & { 0 } \end{array} \right) } , \quad B = { \left( \begin{array} { l l l } { 0 } & { 0 } & { 1 } \\ { 1 } & { 0 } & { 0 } \\ { 0 } & { 1 } & { 0 } \end{array} \right) } , \quad C = { \left( \begin{array} { l l l } { i } & { 0 } & { 0 } \\ { 0 } & { 1 } & { 0 } \\ { 0 } & { 0 } & { 1 } \end{array} \right) }
$$

where $i ^ { 2 } = - 1$ . Here C denotes the complex field.

(a) Compute the order of G.

(b) Find a matrix in G of largest possible order (as an element of G) and compute this order.

(c) Compute the number of elements in G with this largest order.

Problem 6 (2008, 7). (a) Let G be a group of (finite) order n. Show that any irreducible left module over the group algebra CG has complex dimension at least $\sqrt { n }$

(b) Give an example of a group G of order $n \geq 5$ and an irreducible left module over CG of complex dimension $\lfloor { \sqrt { n } } \rfloor$ , the greatest integer to $\sqrt { n }$

Problem 7 (2008, 8). Use the rational canonical form to show that any square matrix M over a field k is similar to its transpose $M ^ { t }$ , recalling that $p ( M ) = 0$ for some $p \in k [ t ]$ if and only if $p ( M ^ { t } ) = 0$

## 12 2007

Problem 1 (2007, 1). Let K be a field of characteristic zero and L a Galois extension of K. Let f be an irreducible polynomial in K[x] of degree 7 and suppose f has no zeroes in L. Show that f is irreducible in $L [ x ]$

Problem 2 (2007, 2). Let K be a field of characteristic zero and $f \in K [ x ]$ an irreducible polynomial of degree n. Let L be a splitting field for f . Let G be the group of automorphisms of L which act trivially on K.

(a) Show that G embeds in the symmetric group $S _ { n }$

(b) For each n, give an example of a field K and polynomial f such that $G = S _ { n }$

(c) What are the possible groups G when n = 3. Justify your answer.

Problem 3 (2007, 3.). Show there are exactly two groups of order 21 up to isomorphism.

## 13 2006

Problem 1 (2006, 2). Let K be the field $\mathbb { Q } ( z )$ of rational functions in a variable z with coefficients in the rational field Q. Let n be a positive integer.
Consider the polynomial $x ^ { n } - z \in K [ x ]$

(a) Show that the polynomial $x ^ { n } - z$ is irreducible over $K .$

(b) Describe the splitting field of $x ^ { n } - z$ over K.

(c) Determine the Galois group of the splitting field of $x ^ { 5 } - z$ over the field K.

Problem 2 (2006, 3.). (a) Let $p \ < \ q \ < \ r$ be prime integers.
Show that a group of order pqr cannot be simple.

(b) Consider groups of orders $2 ^ { 2 } \cdot 3 \cdot p$ where p has the values 5, 7, and 11. For each of those values of $p ,$ either display a simple group of order $2 ^ { 2 } \cdot 3 \cdot p ,$ or show that there cannot be a simple group of that order.

Problem 3 (2006, 4). Let $K / F$ be a finite Galois extension and let $n ~ = ~ [ K : F ]$ There is a theorem (often referred to as the ”normal basis theorem”) which states that there exists an irreducible polynomial $f ( x ) \in F [ x ]$ whose roots form a basis for K as a vector space over F. You may assume that theorem in this problem.

(a) Let $G = \operatorname { G a l } ( K / F )$ . The action of G on K makes K into a finite-dimensional representation space for G over F. Prove that K is isomorphic to the regular representation for G over F.

(The regular representation is defined by letting G act on the group algebra $F [ G ]$ by multiplication on the left.)

(b) Suppose that the Galois group G is cyclic and that F contains a primitive $n ^ { \mathrm { t h } }$ root of unity.
Show that there exists an injective homomorphism $\chi : G \to F ^ { \times }$

(c) Show that K contains a non-zero element a with the following property:

$$
g ( a ) = \chi ( g ) \cdot a
$$

for all $g \in G .$

(d) If a has the property stated in (c), show that $K = F ( a )$ and that $a ^ { n } \in F ^ { \times }$

Problem 4 (2006, 5.). Let G be the group of matrices of the form

$$
\left( \begin{array} { l l l } { 1 } & { a } & { b } \\ { 0 } & { 1 } & { c } \\ { 0 } & { 0 } & { 1 } \end{array} \right)
$$

with entries in the finite field $\mathbb { F } _ { p }$ of $p$ element, where p is a prime.

(a) Prove that G is non-abelian.

(b) Suppose p is odd.
Prove that $g ^ { p } = I _ { 3 }$ for all $g \in G$

(c) Suppose that $p = 2$ It is known that there are exactly two non-abelian groups of order 8, up to isomorphism: the dihedral group $D _ { 8 }$ and the quaternionic group.
Assuming this fact without proof, determine which of these groups G is isomorphic to.

Problem 5 (2006, 7.). There are five nonisomorphic groups of order 8. For each of those groups G, find the smallest positive integer n such that there is an injective homomorphism $\varphi : G \to S _ { n }$

## 14 2005

Problem 1 (2005, 1.). For any group G we define $\Omega ( G )$ to be the image of the group homomorphism $\rho : G \to \operatorname { A u t } ( G )$ where ρ maps $g \in G$ to the conjugation automorphism $x \mapsto g x g ^ { - 1 }$ . Starting with a group $G _ { 0 } ,$ , we define $G _ { 1 } = \Omega ( G _ { 0 } )$ and $G _ { i + 1 } = \Omega ( G _ { i } )$ for all $i \geq 0$ . If $G _ { 0 }$ is of order $p ^ { e }$ for a prime p and integer $e \geq 2 ,$ , prove that $G _ { e - 1 }$ is the trivial group.

Problem 2 (2005, 2.). Let $\mathbb { F } _ { 2 }$ be the field with two elements.

(a) What is the order of $G \mathrm { L } _ { 3 } ( \mathbb { F } _ { 2 } ) ?$

(b) Use the fact that ${ \mathrm { G L } } _ { 3 } ( \mathbb { F } _ { 2 } )$ is a simple group (which you should not prove) to find the number of elements of order 7 in ${ \mathrm { G L } } _ { 3 } ( \mathbb { F } _ { 2 } )$

Problem 3 (2005, 3). Let G be a finite abelian group.
Let $f : \mathbb { Z } ^ { m } \to G$ be a surjection of abelian groups.
We may think of f as a homomorphism of Z-modules.
Let K be the kernel of $f$

(a) Prove that K is isomorphic to $\mathbb { Z } ^ { m }$

(b) We can therefore write the inclusion map $K \to \mathbb { Z } ^ { m }$ as $\mathbb { Z } ^ { m } \to \mathbb { Z } ^ { m }$ and represent it by an $m \times m$ integer matrix A. Prove that | det $A | = | G |$ .

Problem 4 (2005, 4). Let $R = C ( [ 0 , 1 ] )$ be the ring of all continuous real-valued functions on the closed interval [0, 1], and for each $c \in [ 0 , 1 ] ,$ , denote by $M _ { c }$ the set of all functions $f \in R$ such that $f ( c ) = 0$

(a) Prove that $g \in R$ is a unit if and only if $g ( c ) \neq 0$ for all $c \in [ 0 , 1 ]$

(b) Prove that for each $c \in [ 0 , 1 ] , M _ { c }$ is a maximal ideal of R.

(c) Prove that if M is a maximal ideal of T, then $M = M _ { c }$ for some $c \in [ 0 , 1 ]$ . (Hint: compactness of $[ 0 , 1 ]$ may be relevant.)

Problem 5 (2005, 5). Let R and S be commutative rings, and $f : R  S$ a ring homomorphism.

(a) Show that if I is a prime ideal of S, then

$$
f ^ { - 1 } ( I ) = \{ r \in R : f ( r ) \in I \}
$$

is a prime ideal of R.

(b) Let N be the set of nilpotent elements of R:

$$
N = \{ r \in R : r ^ { m } = 0 { \mathrm { ~ f o r ~ s o m e ~ } } m \geq 1 \} .
$$

N is called the nilradical of R. Prove that it is an ideal which is contained in every prime ideal.

(c) Part (a) lets us define a function

f ∗ : {prime ideals of S} → {prime ideals of R}.

$$
I \mapsto f ^ { - 1 } ( I ) .
$$

Let N be the nilradical of R. Show that if $S = R / N$ and $f : R \to R / N$ is the quotient map, then $f ^ { * }$ is a bijection

Problem 6 (2005, 7). Consider the polynomial $f ( x ) = x ^ { 1 0 } + x ^ { 5 } + 1 \in \mathbb { Q } [ x ]$ with splitting field K over Q.

(a) Determine whether $f ( x )$ is irreducible over Q and find $\left[ K : \mathbb { Q } \right]$

(b) Determine the structure of the Galois group $\operatorname { G a l } ( K / \mathbb { Q } )$

Problem 7 (2005, 8). For each prime number p and each positive integer $n ,$ how many elements α are there in $\mathbb { F } _ { p ^ { n } }$ such that $F _ { p } ( \alpha ) = F _ { p ^ { 6 } } ?$