CMI Entrance Exam 2016 Question Paper for M.Sc or PhD Mathematics

CHENNAI MATHEMATICAL INSTITUTE
Postgraduate Programme in Mathematics
MSc/PhD Entrance Examination
18 May 2016
Instructions:
• Enter your Registration Number here: CMI–PG–

• Enter the name of the city where you write this test:

• The allowed time is 3 hours.
• This examination has two parts. You may use the blank pages at the end for your
rough-work.
• Part A is worth 40 marks and will be used for screening. There will be a cut-off for
Part A, which will not be more than twenty (20) marks (out of 40). Your solutions to
the questions in Part B will be marked only if your score in Part A places you over the
cut-off. (In particular, if your score in Part A is at least 20 then your solutions to the
questions in Part B will be marked.) However, note that the scores in both the sections
will be taken into account while making the final decision. Record your answers to
Part A in the attached bubble-sheet.
• Part B is worth 60 marks. You should answer six (6) questions in Part B. In order to
qualify for the PhD Mathematics interview, you must obtain at least fifteen (15) marks
from among the starred questions (17∗ )–(20∗ ). Indicate the six questions to be
marked in the boxes in the bubble-sheet. Write your solutions to Part B in
the page assigned to each question.
• Please read the further instructions given before Part A and inside each part
carefully.

For office use only

Part B Part B (ctd.)

No. Marks Remarks No. Marks Remarks

11 17∗

12 18∗

13 19∗

14 20∗

15

16

Part A

Further remarks: Part B

Total

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CHENNAI MATHEMATICAL INSTITUTE
Postgraduate Programme in Mathematics
MSc/PhD Entrance Examination
18 May 2016

• Registration Number: CMI–PG–
• City where you write this test:

Bubble Sheet
Part A
Part B
In the boxes below, clearly indicate the
1. A B C D
SIX (6) solutions that should be marked.
In order to qualify for the PhD Mathe-
2. A B C D matics interview, you must obtain at least
fifteen (15) marks from among the starred
questions (17∗ )–(20∗ ).
3. A B C D

4. A B C D

5. A B C D

6. A B C D

7. A B C D

8. A B C D

9.

10.

For office use only:

Number of correct answers in Part A:

Marks in Part A:

This page is intentionally left blank.

CHENNAI MATHEMATICAL INSTITUTE
Postgraduate Programme in Mathematics
MSc/PhD Entrance Examination
18 May 2016
Important: Questions in Part A will be used for screening. There will be a cut-off for Part A,
which will not be more than twenty (20) marks (out of 40). Your solutions to the questions in
Part B will be marked only if your score in Part A places you over the cut-off. (In particular, if
your score in Part A is at least 20 then your solutions to the questions in Part B will be marked.)
However, note that the scores in both the sections will be taken into account while making the
final decision. In order to qualify for the PhD Mathematics interview, you must
obtain at least fifteen (15) marks from among the starred questions (17∗ )–(20∗ ).

Notation: N, Z, Q, R and C stand, respectively, for the sets of natural numbers, of integers,
of rational numbers, of real numbers, and of complex numbers. For a prime number p, Fp is
the field with p elements. For a field F , Mm×n (F ) stands for the set of m × n matrices over F
and GLn (F ) is the set of invertible n × n matrices over F . The symbol ı denotes a square-root
of −1.

Part A
Instructions: Each of the questions 1–8 has one or more correct answers. Record your
answers on the attached bubble-sheet by filling in the appropriate circles. Every question is
worth four (4) marks. A solution receives credit if and only if all the correct answers are chosen,
and no incorrect answer is chosen.

(1) We say that two subsets X and Y of R are order-isomorphic if there is a bijective map
φ : X −→ Y such that for every x1 ≤ x2 ∈ X, φ(x1 ) ≤ φ(x2 ), where ‘≤’ denotes the
usual order on R. Choose the correct statement(s) from below:
(A) N and Z are not order-isomorphic;
(B) N and Q are order-isomorphic;
(C) Z and Q are order-isomorphic;
(D) The sets N, Z and Q are pairwise non-order-isomorphic.

(2) Let xn = 1 − n1 sin nπ


3 , n ≥ 1. Write l = lim inf xn and s = lim sup xn . Choose the
correct √statement(s) √from below:
(A) − 23 ≤ l < s ≤ 23 ;
(B) − 21 ≤ l < s ≤ 12 ;
(C) l = −1 and s = 1;
(D) l = s = 0.

(3) Let f : R −→ R be defined as
(
x2 sin( x12 ), if x 6= 0;
f (x) =
0, otherwise.
Choose the correct statement(s) from below:
(A) f is continuous;
(B) f is discontinuous at 0;
(C) f is differentiable;
(D) f is continuously differentiable.
1

(4) Let A ∈ Mm×n (R) be of rank m. Choose the correct statement(s) from below:
(A) The map Rn −→ Rm given by v 7→ Av is injective;  
(B) There exist matrices B ∈ Mm (R) and C ∈ Mn (R) such that BAC = Im | 0n−m ;
(C) There exist matrices B ∈ GLm (R) and C ∈ GLn (R) such that BAC = Im | 0n−m ;
(D) For every (B, C) ∈ Mm (R) × Mn (R) such that BAC = m I | 0 n−m , C is uniquely
determined by B.

(5) Let f : C −→ C be an entire function such that f (z + 1) = f (z + ı) = f (z) for every
z ∈ C. Choose the correct statement(s) from below:
(A) f is constant;
(B) f (z) = 0 for every z ∈ C;
(C) There exist complex numbers a, b such that for every x, y ∈ R, f (x+ıy) = a sin(x)+
ıb cos(y);
(D) f is not necessarily constant but |f (z)| is constant.

(6) What is the cardinality of the centre of O2 (R)? (Definition: The centre of a group G is
{g ∈ G | gh = hg for every h ∈ G}. Hint: Reflection matrices and permutation matrices
are orthogonal.)
(A) 1;
(B) 2;
(C) The cardinality of N;
(D) The cardinality of R.

(7) Let U ⊆ R be a non-empty open subset. Choose the correct statement(s) from below:
(A) U is uncountable;
(B) U contains a closed interval as a proper subset;
(C) U is a countable union of disjoint open intervals;
(D) U contains a convergent sequence of real numbers.

(8) Let R be a commutative ring. The characteristic of R is the smallest positive integer n
such that a + a + · · · + a (n times) is zero for every a ∈ R, if such an integer exists, and
zero, otherwise. Choose the correct statement(s) from below:
(A) For every n ∈ N, there exists a commutative ring whose characteristic is n;
(B) There exists a integral domain with characteristic 57;
(C) The characteristic of a field is either 0 or a prime number;
(D) For every prime number p, every commutative ring of characteristic p contains Fp
as a subring.

Instructions: The answers to questions 9–10 are integers. You are required to write
the answers in decimal form in the attached bubble-sheet. Every question is worth four
(4) marks.

(9) Consider the Q-vector-space
{f : R −→ R | f is continuous and Image(f ) ⊆ Q}.
What is its dimension?

(10) Let p be a prime number and F a field of p23 elements. Let φ : F −→ F be the field
automorphism of F sending a to ap . Let K := {a ∈ F | φ(a) = a}. What is the value
of [K : Fp ]?
2

Part B
Instructions: Answer six (6) questions from below. Provide sufficient justification.
Write your solutions on the page assigned to each question. Each of the questions is worth
ten (10) marks. In order to qualify for the PhD Mathematics interview, you
must obtain at least fifteen (15) marks from among the starred questions
(17∗ )–(20∗ ). Clearly indicate which six questions you would like us to mark in the six
boxes in the bubble sheet. If the boxes are unfilled, we will mark the first six solutions
that appear in your answer-sheet. If you do not want a solution to be considered, clearly
strike it out.

(11) Let U = {(x, y) ∈ R2 | 1 < x2 + y 2 < 4}. Let p, q ∈ U . Show that there is a continuous
map γ : [0, 1] −→ U such that γ(0) = p and γ(1) = q and such that γ is differentiable
on (0, 1).
(12) If I, J are two maximal ideals in a PID that is not a field, then show that IJ is never a
prime ideal.
(13) Let f : C −→ C be an entire function. Suppose that f (z) ∈ R if z is on the real axis or
on the imaginary axis. Show that f 0 (z) = 0 at z = 0.
(14) Let A ⊆ Rn be a closed proper subset. For x, y ∈ Rn , denote the usual (Euclidean)
distance between them by d(x, y). Let x ∈ Rn r A; define δ := inf{d(x, y) | y ∈ A}.
Show that there exists y ∈ A such that δ = d(x, y).
(15) Let F be a field and V a finite-dimensional vector-space over F . Let T : V −→ V be a
linear transformation, such that for every v ∈ V , there exists n ∈ N such that T n (v) = v.
(A) Show that if F = C, then T is diagonalizable.
(B) Show that if char(F ) > 0, then there exists a non-diagonalizable T satisfying the
above hypothesis.
√
(16) Let F = Q(ω, 3 2), where ω ∈ C is a primitive cube-root of unity. Find a Q-basis for
F (with proof). Let µ : F −→ F be the Q-linear map given by µ(a) = ω 2 a. Find the
matrix of µ with respect to the basis obtained above.
(17∗ ) Let G be a non-trivial subgroup of the group (R, +). Show that either G is dense in R
or that G = Z · l where l = inf{x ∈ G | x > 0}.
(18∗ ) Let G be a subgroup of the group of permutations on a finite set X. Let F be the
C-vector-space of all the functions from X to C. G acts on F by (g · f ) : x 7→ f (g −1 (x)).
Show that there is an φ ∈ F such that g · φ = φ for every g ∈ G. Show that there is a
subspace F 0 of F such that F = F 0 ⊕ Chφi and such that g · f ∈ F 0 for every g ∈ G and
f ∈ F 0.
(19∗ ) (A) Let A and B be n × n matrices with entries in N. Show that if B = A−1 then A
and B are permutation matrices. (A permutation matrix is a matrix obtained by
permuting the rows of the identity matrix.)
(B) Let A be an n × n complex matrix that is not a scalar multiple of In . Show that
A is similar to a matrix B such that B1,1 (i.e. the top left entry of B) is 0.

(20∗ ) Let S 1 = {z ∈ C : |z| = 1}. Consider the map Sq : S 1 −→ S 1 ,
Sq(z) = z 2 .
Show that there does not exist a continuous map Sqrt : S 1 −→ S 1 such that Sq ◦ Sqrt =
IdS 1 ? (That is, (Sqrt(w))2 = w.) (Hint: If such a map existed, show that there would
be a bijective continuous map S 1 × {1, −1} −→ S 1 .)
3

Solution to Question (11)

4

Solution to Question (12)

5

Solution to Question (13)

6

Solution to Question (14)

7

Solution to Question (15)

8

Solution to Question (16)

9

Solution to Question (17*)

10

Solution to Question (18*)

11

Solution to Question (19*)

12

Solution to Question (20*)

13