# SOME QUALS PROBLEMS

# SANTIAGO ARANGO-PINEROS ˜

## 1. Algebra

## 1.1. Groups.

(1) Classify the groups of order $1 8 2 = 2 \cdot 7 \cdot 1 3$

(2) Let G be a finite group of order pnm where p is a prime and m is not divisible by p. Prove that if H is a subgroup of G of order $p ^ { k }$ for some $k < n$ , then the normalizer of H in G properly contains H.

(3) Let H be a subgroup of $S _ { n }$ of index n. Prove:

a) There is an isomorphism $f : S _ { n } \to S _ { n }$ such that $f ( H )$ is the subgroup of $S _ { n }$ stabilizing n. In particular, H is isomorphic to $S _ { n }$ −1.

b) The only subgroups of $S _ { n }$ containing H are $S _ { n }$ and H.

(4) a) Prove that a group of order $3 5 1 = 3 ^ { 3 }$ · 13 cannot be simple.

b) Prove that a group of order 33 must be cyclic.

(5) a) Let G be a group, and Z(G) the center of G. Prove that if $G / Z ( G )$ is cyclic, then G is abelian.

b) Prove that a group of order $p ^ { n }$ , where p is a prime and $n \geq 1$ has non-trivial center.

c) Prove that a group of order $p ^ { 2 }$ must be abelian.

(6) Let G be a finite group.

a) Prove that if $H < G$ is a proper subgroup, then G is not the union of conjugates of H.

b) Suppose that G acts transitively on a set X with $| X | > 1$ . Prove that there exists an element of G with no fixed points in X.

(7) Classify all groups of order 15 and of order 30.

(8) Count the number of p-Sylow subgroups of $S _ { p }$

(9) a) Let G be a group of order n. Suppose that for every divisor d of $n , G$ contains at most one subgroup of order d. Show that G is clyclic.

b) Let F be a field.
Show that every finite subgroup of the group of units $F ^ { \times }$ is cyclic.

## 1.2. Fields and Galois Theory.

(1) Let K and L be finite fields.
Show that K is contained in L if and only if # $K = p ^ { r }$ and # $L = p ^ { s }$ for the same prime $p ,$ and $r \leq s .$

(2) Let K and L be finite fields with $K \subseteq L$ . Prove that L is Galois over K and that $\operatorname { G a l } ( L / K )$ is cyclic.

(3) Fix a field F , a separable polynomial $f \in F [ x ]$ of degree $n \geq 3$ , and a splitting field L for f . Prove that if $[ L : F ] =$ n! then:

a) f is irreducible.

b) For each root r of $f , r$ is the unique root of f in $F ( r )$

c) For every root r of f, there are no proper intermediate fields $F \subset L \subset F ( r )$

(4) a) Show that $\sqrt { 2 + { \sqrt { 2 } } }$ is a root of $p ( x ) = x ^ { 2 } - 4 x ^ { 2 } + 2 \in \mathbf { Q } [ x ]$

b) Prove that $\mathbf { Q } ( { \sqrt { 2 + { \sqrt { 2 } } } } )$ is a Galois extension of Q and find its Galois group.
(Hint: note that $\sqrt { 2 - { \sqrt { 2 } } }$ is another root of $p ( x ) )$

c) Let $f ( x ) = x ^ { 3 } - 5$ . Determine the splitting field K of $f ( x )$ over Q and the Galois group of $f ( x )$ . Give an example of a proper sub-extension $\mathbf { Q } \subset L \subset K$ , such that $L / \mathbf { Q }$ is Galois.

## 1.3. Rings.

(1) An integral domain R is said to be an Euclidean domain if there is a function $N : R \to \{ n \in \mathbf { Z } \mid n \geq 0 \}$ such that $N ( 0 ) = 0$ and for each $a , b \in R$ with $b \neq 0$ , there exist elements $q , r \in R$ with

$$
a = q b + r , \quad \mathrm { a n d } \quad r = 0 \mathrm { o r } N ( r ) < N ( b ) .
$$

Prove:

a) The ring $F [ [ x ] ]$ of power series over a field F is an Euclidean domain.

b) Every Euclidean domain is a PID.

(2) Let F be a field, and let R be the subring of $F [ X ]$ of polynomials with X coefficient equal to 0. Prove that R is not a UFD.

(3) R is a commutative ring with 1. Prove that if I is a maximal ideal in R, then $R / I$ is a field.
Prove that if R is a PID, then every nonzero prime ideal in R is maximal.
Conclude that if R is a PID and $p \in R$ is prime, then $R / ( p )$ is a field.

## 1.4. Linear Algebra.

(1) Prove that any square matrix is conjugate to its transpose matrix.
(You may prove it over C).

(2) Determine the number of conjugacy classes of $1 6 \times 1 6$ matrices with entries in Q and minimal polynomial $( x ^ { 2 } + 1 ) ^ { 2 } ( x ^ { 3 } + 2 ) ^ { 2 }$

(3) Let V be a vector space over a field F . The evaluation map $e \colon V \to$ $( V ^ { \vee } ) ^ { \vee }$ is defined by $e ( v ) ( f ) : = f ( v )$ for $v \in V$ and $f \in V ^ { \vee }$

a) Prove that e is an injection.

b) Prove that e is an isomorphism if and only if V is finite dimensional.

(4) Let R be a principal ideal domain that is not a field, and write F for its field of fractions.
Prove that F is not a finitely generated R-module.

(5) Carefully state Zorn’s lemma and use it to prove that every vector space has a basis.

## 2. Analysis

## 2.1. Complex Analysis.

(1) Use residues to compute the integral

$$
\int _ { 0 } ^ { \infty } \frac { \cos { x } } { ( x ^ { 2 } + 1 ) ^ { 2 } } \mathrm { d } x .
$$

(2) State and prove the Cauchy integral formula for holomorphic functions.

(3) Use the Cauchy integral formula to prove the maximal principle for analytic functions.

(4) Let f be an entire function and suppose that $| f ( z ) | \leq A | z | ^ { 2 }$ for all z and some constant A. Show that $f$ is a polynomial of degree $\leq 2$

(5) a) State the Schwarz lemma for analytic functions in the unit disc.

b) Let $f : \mathbf { D } $ D be an analytic map from the unit disc D into itself.
Use the Schwarz lemma to show that for each $a \in \mathbf { D }$ we have

$$
{ \frac { | f ^ { \prime } ( a ) | } { 1 - | f ( a ) | ^ { 2 } } } \leq { \frac { 1 } { 1 - | a | ^ { 2 } } } .
$$

(6) State the Riemann mapping theorem and prove the uniqueness part.

(7) Compute the integrals

$$
\int _ { | z - 2 | = 1 } \frac { e ^ { z } } { z ( z - 1 ) ^ { 2 } } \mathrm { d } z , \quad \int _ { 0 } ^ { \infty } \frac { \cos 2 x } { x ^ { 2 } + 2 } \mathrm { d } x .
$$

(8) Let $\left( f _ { n } \right)$ be a sequence of holomorphic functions in a domain D. Suppose that $f _ { n }  f$ uniformly on each compact subset of $D .$ Show that

a) $f$ is holomorphic on D.

b) $f _ { n } ^ { \prime } \to f ^ { \prime }$ uniformly on each compact subset of D.

(9) If $f$ is a non-constant entire function, then $f ( \mathbf { C } )$ is dense in the plane.

(10) a) State Rouche’s theorem.

b) Let $f$ be analytic in a neighborhood of 0, and satisfying $f ^ { \prime } ( 0 ) \neq$ 0. Use Rouche’s theorem to show that there exists a neighborhood $U$ of 0 such that $f$ is a bijection in $U$

(11) Let $f$ be a meromorphic function in the plane such that

$$
\operatorname* { l i m } _ { | z | \to \infty } | f ( z ) | = \infty .
$$

a) Show that f has only finitely many poles.

b) Show that f is a rational function.

## 2.2. Real Analysis.

(1) Describe the process that extends a measure on an algebra A of subsets of X, to a complete measure defined on a σ-algebra B containing A. State the corresponding definitions and results (without proofs).

(2) State and prove Fatou’s Lemma on a general measurable space.

(3) a) State the Dominated Convergence Theorem for Lebesgue integrals.

b) Let $\left\{ f _ { n } \right\}$ be a sequence of measurable functions on a Lebesgue measurable set E which converges in measure to a function f on E. Suppose that for every n, $| f _ { n } | \le g$ with g integrable on E. Using the above theorem show that

$$
\int _ { E } | f _ { n } - f | \longrightarrow 0 .
$$

(4) Let $f \in L ^ { 1 } ( [ 0 , 1 ] )$ . Show that

a) The limit li $\mathrm { m } _ { p \to 0 ^ { + } } \| f \| _ { p }$ exists.

b) If $m \{ x : f ( x ) = 0 \} > 0$ , then the above limit is zero.

(5) Let f be a continuous function on [0, 1]. Show that the following statements are equivalent.

a) f is absolutely continuous.

b) For any $\epsilon > 0$ there exists $\delta > 0$ such that $m ( f ( E ) ) < \epsilon$ for any set $E \subseteq [ 0 , 1 ]$ with $m ( E ) < \delta$

c) $m ( f ( { \cal E } ) ) = 0$ for any set $E \subseteq [ 0 , 1 ]$ with $m ( E ) = 0$