CMI Entrance Exam 2023 Question Paper M.Sc PhD Mathematics

CHENNAI MATHEMATICAL INSTITUTE
Postgraduate Programme in Mathematics
MSc/PhD Entrance Examination
7th May 2023

Part A
(1) Let 𝑅 be an integral domain containing C such that it is a finite-dimensional C-vector-space.
Pick the correct statement(s) from below.
(A) For every 𝑎 ∈ 𝑅, the set {1, 𝑎, 𝑎 2, . . .} is linearly dependent over C.
(B) 𝑅 is a field.
(C) 𝑅 = C.
(D) The transcendence degree of 𝑅 over C is 1.
(2) Let 𝑅 be a euclidean domain that is not a field. Let 𝑑 : 𝑅 \{0} −→ N be the euclidean size (degree)
function. Write 𝑅 × for the invertible elements of 𝑅. Pick the correct statements from below.
(A) 𝑅 = 𝑅 × ∪ {0}.
(B) There exists 𝑎 ∈ 𝑅 \ (𝑅 × ∪ {0}) such that 𝑑 (𝑎) = inf {𝑑 (𝑟 ) | 𝑟 ∈ 𝑅 \ (𝑅 × ∪ {0})}.
(C) With 𝑎 defined as above, for all 𝑟 ∈ 𝑅, there exists 𝑢 ∈ 𝑅 × ∪ {0} such that 𝑎 divides (𝑟 − 𝑢).
(D) With 𝑎 defined as above, the ideal generated by 𝑎 is a maximal ideal.
(3) Let 𝑋 be a compact topological space. Let 𝑓 : 𝑋 −→ R be a function satisfying 𝑓 −1 ( [𝑛, ∞)) is
closed for all 𝑛 ∈ N. Pick the correct statements from below.
(A) 𝑓 is continuous.
(B) 𝑓 (𝑈 ) is open for each open subset 𝑈 of 𝑋 .
(C) 𝑓 (𝑈 ) is closed for each closed subset 𝑈 of 𝑋 .
(D) 𝑓 is bounded above.
(4) Let 𝑓 : [0, 1] −→ R be a continuous function and 𝐸 ⊆ [0, 1]. Which of the following are true?
(A) If 𝐸 is closed, then 𝑓 (𝐸) is closed.
(B) If 𝐸 is open, then 𝑓 (𝐸) is open.
(C) If 𝐸 is a countable union of closed sets, then 𝑓 (𝐸) is a countable union of closed sets.
(D) If 𝑓 injective and 𝐸 is a countable intersection of open sets, then 𝑓 (𝐸) is a countable inter-
section of open sets.
(5) Consider the real matrix  
𝜆 2
𝐴= .
3 5
Assume that −1 is an eigenvalue of 𝐴. Which of the following are true?
(A) The other eigenvalue is in C \ R.
(B) 𝐴 + 𝐼 2 is singular.
(C) 𝜆 = 1.
(D) Trace of 𝐴 is 5.
(6) Let 𝑎𝑛 , 𝑛 ≥ 1, be a sequence of positive real numbers such that 𝑎𝑛 −→ ∞ as 𝑛 −→ ∞. Then which
of the following are true?
(A) There exists a natural number 𝑀 such that
∞
∑︁ 1
𝑛=1
(𝑎𝑛 ) 𝑀
(B)
∞
∑︁ 1
𝑛=1
(𝑛 2𝑎𝑛 )
(C)
∞
∑︁ 1
𝑛=1
(𝑛𝑎𝑛 )
1

(D) For all positive real numbers 𝑅,
∞
∑︁ 𝑅𝑛
𝑛=1
(𝑎𝑛 )𝑛
(7) Let 𝐴 be the ring of all entire functions under point-wise addition and multiplication. Then
which of the following are true?
(A) 𝐴 does not have non-zero nilpotent elements.
(B) In the group of the units of 𝐴 (under multiplication), every element other than 1 has infinite
order.
(C) For every 𝑓 ∈ 𝐴, there is a sequence of polynomials which converges to 𝑓 uniformly on
compact sets.
(D) The ideal generated by 𝑧 and sin 𝑧 is principal.
(8) Which of the following groups are cyclic?
(A) Z/2Z ⊕ Z/9Z
(B) Z/3Z ⊕ Z/9Z
(C) Every group of order 18.
(D) (Q× , ·)
(9) Let 𝑝, 𝑞 be distinct prime numbers and let 𝜁𝑝 , 𝜁𝑞 denote (any) primitive 𝑝-th and 𝑞-th roots of
unity, respectively. Choose all the correct statements.
(A) 𝜁 13 ∉ Q(𝜁 31 ).
(B) If 𝑝 divides 𝑞 − 1, then 𝜁𝑝 ∈ Q(𝜁𝑞 ).
(C) If 𝜁𝑝 ∈ Q(𝜁𝑞 ), then 𝑝 − 1 divides 𝑞 − 1.
(D) If there exists a field homomorphism Q(𝜁𝑝 ) −→ Q(𝜁𝑞 ), then 𝑝 − 1 divides 𝑞 − 1.
(10) Let 𝑓 , 𝑔 : R2 −→ R be functions. Let 𝐹 = (𝑓 , 𝑔) : R2 −→ R2 . Assume that 𝐹 is infinitely
differentiable and that 𝐹 (0, 0) = (0, 0). Suppose further that the function 𝑓 𝑔 : R2 −→ R is
everywhere non-negative. Then
(A) 𝑓𝑥 (0, 0) = 0, 𝑓𝑦 (0, 0) = 0.
(B) 𝑔𝑥 (0, 0) = 0, 𝑔𝑦 (0, 0) = 0.
(C) The image of 𝐹 is not dense in R2 .
(D) det 𝐽 (0, 0) = 0 where 𝐽 is the matrix of first partial derivatives (i.e., the jacobian matrix).

Part B
(11) Let 𝑓 : R ≥0 −→ R be the function
(
1, 𝑥 =0
𝑓 (𝑥) = −𝑥
𝑥 , 𝑥>0
Determine whether the following statement is true:
∫ 1 ∞
∑︁
𝑓 (𝑥)d𝑥 = 𝑛 −𝑛 .
0 𝑖=0

(12) (A) (3 marks) Let 𝐺 be a group such that |𝐺 | = 𝑝 𝑎𝑑 with 𝑎 ≥ 1 and (𝑝, 𝑑) = 1. Let 𝑃 be a Sylow
𝑝-subgroup and let 𝑄 be any 𝑝-subgroup of 𝐺 such that 𝑄 is not a subgroup of 𝑃. Show that
𝑃𝑄 is not a subgroup of 𝐺.
(B) (7 marks) Let Γ be a group that is the direct product of its Sylow subgroups. Show that every
subgroup of Γ also satisfies the same property.
(13) (A) (5 marks) Let 𝑛 ≥ 2 be an integer. Let 𝑉 be the R-vector-space of homogeneous real polyno-
mials in three variables 𝑋, 𝑌 , 𝑍 of degree 𝑛. Let 𝑝 = (1, 0, 0). Let
𝜕𝑓
𝑊 = {𝑓 ∈ 𝑉 | 𝑓 (𝑝) = (𝑝)}
𝜕𝑋
Determine the dimension of 𝑉 /𝑊 .
2

(B) (5 marks) A linear transformation 𝑇 : R9 −→ R9 is defined on the standard basis 𝑒 1, . . . , 𝑒 9
by
𝑇 𝑒𝑖 = 𝑒𝑖 −1, 𝑖 = 3, . . . , 9
𝑇 𝑒2 = 𝑒3
𝑇 𝑒1 = 𝑒1 + 𝑒3 + 𝑒8 .
Determine the nullity of 𝑇 .
(14) Let 𝐹 be a field and 𝑅 a subring of 𝐹 that is not a field. Let 𝑥 be a variable. Let 𝑆 = {𝑎 0 + 𝑎 1𝑥 + · · · +
𝑎𝑛 𝑥 𝑛 | 𝑛 ≥ 0 and 𝑎 0 ∈ 𝑅, 𝑎 1, · · · 𝑎𝑛 , ∈ 𝐹 }.
(A) (2 marks) Show that, with the natural operations of addition and multiplication of polyno-
mials, 𝑆 is an integral domain.
(B) (4 marks) Let 𝐼 = {𝑓 (𝑥) ∈ 𝑆 |𝑓 (0) = 0}. Determine whether 𝐼 is a prime ideal.
(C) (4 marks) Determine whether 𝑆 is a PID.
(15) (A) (6 marks) Let 𝑓 , 𝑔 : [0, 1] ↦→ R be monotonically increasing continuous functions. Show
that ∫ 1  ∫ 1  ∫ 1
𝑓 (𝑥)𝑑𝑥 𝑔(𝑥)𝑑𝑥 ≤ 𝑓 (𝑥)𝑔(𝑥)𝑑𝑥 .
0 0 0
(Hint: try double integrals.)
(B) (4 marks) Let 𝑓 : R −→ R be an infinitely differentiable function such that 𝑓 (1) = 𝑓 (0) = 0.
Also, suppose that for some 𝑛 > 0, the first 𝑛 derivatives of 𝑓 vanish at zero. Then prove that
for the (𝑛 + 1)th derivative of 𝑓 , 𝑓 (𝑛+1) (𝑥) = 0 for some 𝑥 ∈ (0, 1).
(16) (A) (5 marks) Consider the eucliean space R𝑛 with the usual norm and dot product. Let x, y ∈ R𝑛
be such that
∥x + 𝑡y∥ ≥ ∥x∥, for all 𝑡 ∈ R.
Show that x · y = 0.
(B) (5 marks ) Consider the vector field 𝑣® = (𝑣 𝑥 , 𝑣 𝑦 ) (with components (𝑣 𝑥 , 𝑣 𝑦 )) on R2 :
𝑣 𝑥 (𝑥, 𝑦) = 𝑥 − 𝑦, 𝑣 𝑦 (𝑥, 𝑦) = 𝑦 + 𝑥
Compute the line integral of 𝑣® along the unit circle (counterclockwise). Is there a function 𝑓
such that 𝑣® = grad𝑓 ?
(17∗ ) Denote by 𝑉 the Q-vector-space Q[𝑋 ] (polynomial ring in one variable 𝑋 ). Show that 𝑉 ∗ is not
isomorphic to 𝑉 .
(18∗ ) Let 𝑓 be a non-constant entire function with 𝑓 (0) = 0. Let 𝑢 and 𝑣 be the real and imaginary
parts of 𝑓 respectively. Let 𝑅 > 0 and
𝐵 = sup{𝑢 (𝑧) : |𝑧| = 𝑅}.
(A) (2 marks) Show that 𝐵 > 0.
(B) (2 marks) Consider the function
𝑓 (𝑧)
𝐹 (𝑧) := .
𝑧 (2𝐵 − 𝑓 (𝑧))
Show that 𝐹 is analytic on the open ball with radius 𝑅 and continuous on the boundary {𝑧 :
|𝑧| = 𝑅}.
(C) (3 marks) Show that sup{|𝐹 (𝑧)| : |𝑧| = 𝑅} ≤ 𝑅1 .
(D) (3 marks) Show that  
𝑅
sup |𝑓 (𝑧)| : |𝑧| = ≤ 2𝐵.
2
(19∗ ) Let 𝑈 (𝑛) be the group of 𝑛 × 𝑛 unitary complex matrices. Let 𝑃 ⊂ 𝑈 (𝑛) be the set of all finite
order elements of 𝑈 (𝑛), that is, 𝑃 = {𝑋 ∈ 𝑈 (𝑛) | 𝑋 𝑚 = 1 for some 𝑚 ≥ 1}. Show that 𝑃 is dense
in 𝑈 (𝑛).
(20∗ ) Let 𝐴 be a non-trivial subgroup of R generated by finitely many elements. Let 𝑟 be a real number
such that 𝑥 −→ 𝑟𝑥 is an automorphism of 𝐴. Show that 𝑟 and 𝑟 −1 are zeros of monic polynomials
with integer coefficients.
3