CMI Entrance Exam 2024 Question Paper M.Sc PhD Mathematics

Entrance Exam
2024

QUESTION
PAPER

CHENNAI MATHEMATICAL INSTITUTE
Postgraduate Programme in Mathematics
MSc/PhD Entrance Examination
19th May 2024

Part A
7 3
(1) Let f (z) = z − 4z − 11. Pick the correct statement(s) from below.
(A) f (z) has at least 1 zero in the open set {|z| > 2}.
(B) f (z) has at least 5 zeroes in the annular region {1 < |z| < 2}.
(C) f (z) has exactly 6 zeroes in the annular region {1 < |z| < 2}.
(D) f (z) has exactly 1 zero in the closed disc {|z| ≤ 1}.
(2) A region in C is a non-empty open connected set. Select all the statement(s) that are true.
(A) Let f be a function on a region Ω such that the integral of f along the boundary of any closed triangle in
Ω is zero. Then f is analytic on Ω.
(B) There exist
a region Ω containing the real interval (0, 1) and a non-zero analytic function f : Ω → C such
that f n1 = 0 for all positive integers n.
(C) Let f be an analytic function on C \ {0} with an essential singularity at z = 0. Then lim |f (z)| = ∞.
z→0
(D) Every bounded analytic function on C \ {0} is constant.
(3) Let u and v be real-valued functions on R2 defined as follows:
( 3
x −3xy 2
x2 +y 2 if (x, y) ̸= (0, 0),
u(x, y) =
0 otherwise.
( 3 2
y −3yx
x2 +y 2 if (x, y) ̸= (0, 0),
v(x, y) =
0 otherwise.
Let f : R2 −→ R2 be the function f (x, y) = (u(x, y), v(x, y)). Pick the correct statement(s) from below.
(A) ∂u ∂u ∂v ∂v
∂x , ∂y , ∂x and ∂y exist at (0, 0).
(B) ∂u
∂x is continuous at (0, 0).
(C) For every fixed (a, b) ̸= (0, 0) ∈ R2 , the function t 7→ f (ta, tb) is a differentiable function (of t).
(D) f is differentiable at (0, 0).
(4) Let G (respectively, H) be a Sylow 2-subgroup (respectively, Sylow 7-subgroup) of the symmetric group S17 .
Pick the correct statement(s) from below.
(A) The order of G is 215 .
(B) H is abelian.
(C) G has a subgroup isomorphic to Z/8Z × Z/8Z.
(D) If σ ∈ S17 has order 4, then σ is a 4-cycle.
(5) Let p ≥ 3 be a prime number and V be an n-dimensional vector space over Fp . Let T : V → V be a linear
transformation. Select all the true statement(s) from below.
(A) T has an eigenvalue in Fp .
(B) If T p−1 = I, then the minimal polynomial of T has distinct roots in Fp .
(C) If T ̸= I and T p−1 = I, then the characteristic polynomial of T has distinct roots in Fp .
(D) If T p−1 = I, then T is diagonalizable over Fp .
(6) Let X be a subset of R3 . We say that X has property S if it contains at least two elements and every Cauchy
sequence in X has a limit point in X. Pick the correct statement(s) from below.
(A) If X has property S then it must be compact.
(B) If X has property S then it must be closed.
(C) Suppose that X has property S and it further satisfies the following condition: if (a1 , b1 , c1 ), (a2 , b2 , c2 ) ∈
X, then (a1 + a2 , b1 + b2 , c1 +, c2 ) ∈ X. Then X is dense in R3 .
(D) Suppose that X has property S and it further satisfies the following condition: if (a, b, c) ∈ X, then
( a2 , 2b , 2c ) ∈ X. Then X is dense in R3 .
(7) Let f be a continuous real-valued function on [0, 1] such that
Z 1 Z 1
f (x)dx = xf (x)dx = 0.
0 0
Pick the correct statement(s) from below.
1

(A) f must have a zero in [0, 1].
(B) f has at least two zeros, counted with multiplicity, in [0, 1].
(C) If f ̸≡ 0, then f has exactly two zeros in [0, 1].
(D) f ≡ 0.
(8) Which of the following statement(s) are true?
(A) Every prime ideal of a finite commutative ring with unity is maximal.
(B) A commutative ring with unity whose set of all ideals is countably infinite is necessarily a countable ring.
(C) Let R be a unique factorisation domain and K be its field of fractions. There exists an irreducible element
α ∈ R and an element β ∈ K such that β 2 = α.
(D) Every subring R (with unity) of Q with Z ⫋ R ⫋ Q has infinitely many prime ideals.
(9) Which of the following statement(s) are true?
(A) If F1 , F2 are finite field extensions of Q such that [F1 : Q] = [F2 : Q], then F1 , F2 are isomorphic as
fields.
(B) If F1 , F2 are finite field extensions of R such that [F1 : R] = [F2 : R], then F1 , F2 are isomorphic as
fields.
(C) Let F be a finite field. If F1 , F2 be finite field extensions of F such that [F1 : F] = [F2 : F], then F1 , F2
are isomorphic as fields. √ √
(D) Let ω ∈ C be a primitive cube root of unity and let 3 2 ∈ R be a cube root of 2. Let K = Q(ω, 3 2). If
F1 , F2 are subfields of K such that [K : F1 ] = [K : F2 ] = 2, then F1 = F2 .
(10) Let X = {α ∈ C | α satisfies a monic polynomial over Q}. (I.e., X is the algebraic closure of Q in C.)
Endow X with the subspace topology of the euclidean metric topology from C. Pick the correct statement(s)
from below.
(A) X is closed.
(B) X is complete.
(C) X is unbounded.
(D) X is connected.

Part B
(11) Let n ≥ 3 be an integer. Write D2n for the dihedral group with 2n elements. Show that the automorphism
group of D2n has at most nφ(n) elements. (Here φ(n) is the number of positive integers that are relatively
prime to n.)
(12) Let SL(2, R) be the group of 2 × 2 matrices with real entries and determinant 1, endowed with the subspace
topology of R4 . Consider the continuous map f : SL(2, R) −→ C given by
 
a b ai + b
f = .
c d ci + d

(A) (4 marks) Show that f maps SL(2, R) onto the upper half plane H = {z : Im(z) > 0}.
(B) (6 marks) Assume the following two facts:
(i) For all M, N ∈ SL(2, R), f (M ) = f (N ) if and only if M −1 N is an orthogonal matrix.
(ii) The map f is an open map.
Now show that for every compact K ⊆ H, f −1 (K) is compact.
(13) Let G be a finite group of odd order and 1 < d < |G| be a divisor of |G|. Assume that G has exactly three
subgroups H1 , H2 and H3 of order d. Suppose that H1 is not normal in G. For each i = 1, 2, 3, let Ni denote
the normalizer of Hi . Let S := {H1 , H2 , H3 }.
(4 marks) For each g ∈ G let sg denote the cardinality of the set {H ∈ S | gHg −1 = H}. Show that
(A) P
g∈G sg = |N1 | + |N2 | + |N3 |.
(B) (6 marks) Show that G ̸= N1 ∪ N2 ∪ N3 .
(14) (A) (5 marks) Let X be a non-empty finite set and let R be the ring of Z-valued functions on X, with pointwise
addition and multiplication. Let S be an additive subgroup of R such that the multiplicative identity 1R ̸∈
S and such that for all s, s′ ∈ S, ss′ ∈ S. Show that there exists x ∈ X and a prime number p such that
f (x) is divisible by p for all f ∈ S (Hint: Consider the sets {f (x) : f ∈ S} for all x ∈ X.)
(B) (5 marks) Let K be a subfield of C with [K : Q] = 2. Let P ∈ Q[x] be irreducible over Q. Show that P
is either irreducible in K[x] or splits as product of two irreducible polynomials in K[x].
(15) Let A(X), B(X) be non-zero polynomials in C[X] such that 0 ≤ deg A ≤ deg B − 2 and A(X) and B(X)
do not share any roots. Let α1 , α2 , . . . , αk be the roots of B(X). Suppose that each of them is a simple root.
2

Show that
k
X A(αj )
= 0.
j=1
B ′ (αj )
(16) (A) (3 marks) Let f : R −→ R be a continuous function such that f (r) = f (r + n1 ) for each r ∈ Q and each
positive integer n. Prove or disprove the following statement: f is a constant function.
(B) (7 marks) Let an , n ≥ 1 be a sequence of non-negative real numbers such that am+n ≤ am + an for all
m, n. Show that
an na
n
o
lim = inf |n≥1 .
n−→∞ n n
(17∗ ) Let p be a prime number. Let n ≥ 2 be an integer. Let V be an n-dimensional Fp -vector space. Determine, with
a proof, the number of two-dimensional Fp -subspaces of V .
(18∗ ) Let R be the ring of all the real-valued functions on N × N. Show that R contains a subring isomorphic to the
polynomial ring R[X, Y ].
2
(19∗ ) Let Mn (R) be the space of n × n matrices with real entries, identified with the euclidean space Rn . Let X be
a compact subset of Mn (R), and S ⊂ C be the set of all eigenvalues of the matrices in X. Show that S is a
compact subset of C.
2
∗
(20 ) Fix 0 < λ < 1. Choose ϵ > 0 such that ϵ + λ4 ≤ λ2 . Consider the metric space
λ
X := {ψ ∈ C 1 ([−ϵ, ϵ]) : |y + ψ(y)| ≤ for all y ∈ [−ϵ, ϵ]},
2
with the induced supremum metric from C 1 ([−ϵ, ϵ]), which we denote by d. (Recall: C 1 ([−ϵ, ϵ]) is the set of
real-valued differentiable functions on [−ϵ, ϵ] whose derivative is continuous.)
(A) (1 mark) Show that there is a function A : X −→ X given by
(Aψ)(y) = −(y + ψ(y))2 .
(B) (2 marks) Show that if ψ ∈ X is such that Aψ = ψ, then the function x = y + ψ(y) is an inverse to the
function y = x + x2 , locally near the origin.
In the next few steps, we show that such a ψ exists.
(C) (2 marks) Show that d(Aψ1 , Aψ2 ) ≤ λd(ψ1 , ψ2 ).
(D) (4 marks) Let ϕ ∈ X. Show that the sequence An ϕ, n ≥ 1 is a Cauchy sequence, and it has a limit. By An ,
we mean the n-fold composition A ◦ A ◦ · · · ◦ A. (You may use the fact that X is complete with respect to
d.)
(E) (1 mark) Show that there exists ψ ∈ X such that Aψ = ψ.

3

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