## FALL 2007 PRELIMINARY EXAMINATION

1A. Let Z[i] be the set of complex numbers of the form $a + b i$ where a and b range over all integers. List all subrings of $\mathbb { Z } [ i ]$ . (Your list should contain each subring exactly once.)

2A. Let $f ( z )$ and $g ( z )$ be entire functions such that $f ^ { \prime } ( z ) ~ = ~ g ( z ) , ~ g ^ { \prime } ( z ) ~ = ~ - f ( z )$ , and $f ( 2 z ) = 2 f ( z ) g ( z )$ for all $z \in \mathbb { C }$ . Find all possibilities for f (z).

3A. Let A be an $n \times n$ Hermitian matrix, and let $x \in \mathbb { C } ^ { n }$ be a vector such that $A ^ { 2 } x = 0$ Prove that $A x = 0$

4A. Let $( a _ { n } ) _ { n \geq 1 }$ and $( b _ { n } ) _ { n \geq 1 }$ be sequences of real numbers. Suppose that $0 \leq a _ { n + 1 } \leq a _ { n } + b _ { n }$ for all $n \geq 1$ , and that $\sum _ { n = 1 } ^ { \overline { { \infty } } } b _ { r }$ converges. Prove that $\operatorname* { l i m } _ { n \to \infty } a _ { n }$ exists and is finite.

5A. Suppose that G is a finite group such that for each subgroup H of G there exists a homomorphism $\phi \colon G \to H$ such that $\phi ( h ) = h$ for all $h \in H$ . Show that G is a product of groups of prime order.

6A. Let $f ( z ) = z ^ { 4 } + \frac { z ^ { 3 } } { 4 } - \frac { 1 } { 4 }$ . How many zeros does f have in $\{ z \in \mathbb { C } : \frac { 1 } { 2 } < | z | < 1 \} ?$

7A. Let $P \in \mathbb { R } ^ { n \times n }$ be a matrix satisfying $P ^ { 3 } = P$ . Let r be the rank of P and assume $r > 0$ Show that there exist matrices $U , V \in \mathbb { R } ^ { n \times r }$ satisfying $V ^ { T } U = I _ { r }$ such that

$$
P = U S V ^ { T } ,
$$

where $I _ { r }$ is the $r \times r$ identity matrix, and S is an $r \times r$ diagonal matrix with ±1’s on the diagonal.

8A. Suppose that $( b _ { n } ) _ { n \geq 1 }$ is a sequence of positive real numbers tending to infinity such that $b _ { n } / n \to 0$ . Must there exist a sequence $( a _ { n } ) _ { n \geq 1 }$ such that $( a _ { 1 } + \cdot \cdot \cdot + a _ { n } ) / n \to 0$ and lim $\textstyle \operatorname* { s u p } _ { n \to \infty } ( a _ { n } / b _ { n } ) = \infty ?$

9A. Let G be a non-abelian group of order 16 having a subgroup H isomorphic to $C _ { 2 } { \times } C _ { 2 } { \times } C _ { 2 }$ (where $C _ { 2 }$ denotes a cyclic group of order 2). Prove that the number of elements of G of exact order 2 is either 7 or 11.

1B. Let $f ( z )$ be a polynomial with complex coefficients, and let a be a complex number.   
Prove that $\{ a , f ( a ) , f ( f ( a ) ) , \ldots \}$ is not dense in C.

2B. Let A be an $n \times n$ complex matrix. Suppose that m is a positive integer such that $A ^ { m }$ is diagonalizable. Prove that $A ^ { m + 1 }$ is diagonalizable.

3B. Let $\{ u _ { 1 } , u _ { 2 } , \cdots , u _ { k } \}$ be a set of linearly independent vectors in $\mathbb { R } ^ { n }$ , and let A be a closed set in Rk. Let S be the set of linear combinations $\alpha _ { 1 } u _ { 1 } + \alpha _ { 2 } u _ { 2 } + \cdot \cdot \cdot + \alpha _ { k } u _ { k }$ obtained as $\left( \alpha _ { 1 } , \alpha _ { 2 } , \ldots , \alpha _ { k } \right)$ ranges over all points of A. Show that S is a closed subset of $\mathbb { R } ^ { n }$

4B. Let K and L be fields, and let $K \times L$ be the product ring, with addition and multiplication defined componentwise. Find all prime ideals of $K \times L$

5B. Let $f ( z )$ be an entire function and let $a _ { 1 } , \ldots , a _ { n }$ be all zeros of $f$ in $\mathbb { C } .$ . Suppose that there exist real numbers $R > 0$ and $a > 1$ such that $| f ( z ) | \geq | z | ^ { a }$ for all $| z | \geq R$ . Prove that

$$
\sum _ { j = 1 } ^ { n } \operatorname { R e s } _ { z = a _ { j } } { \frac { 1 } { f ( z ) } } = 0 .
$$

6B. Given a positive integer n, what are the possible values of the triple $( \operatorname { r k } ( A ) , \operatorname { r k } ( B ) , \operatorname { r k } ( C ) )$ as $A , B , C$ range over real $n \times n$ matrices satisfying $A + B + C = 0 ?$

7B. Let f be continuous on $[ 0 , \infty )$ and suppose that $\operatorname* { l i m } _ { x \to \infty } f ( x )$ exists and is finite. Must $f$ be uniformly continuous? Give a proof or a counterexample.

8B. Show that for every positive integer n, there exists an irreducible polynomial over $\mathbb { Q }$ of degree n such that all its roots are real.

9B. Let f be holomorphic on a neighborhood of the closed disk $\overline { { B _ { 1 } ( 0 ) } } = \{ z : | z | \leq 1 \}$ Suppose that $\mathrm { m a x } _ { | z | = 1 } | f ( z ) | \leq 1$ . Prove that there exists a complex number z such that $| z | \le 1$ and $f ( z ) = { \dot { z } }$