CSIR NET Maths previous year question papers

Single choice · Rings and Fields

1. A

   The set {a ∈ A : 1/a ∉ A} ∪ {0} is an additive subgroup of $\mathbb{Q}$.

2. B

   A has at most one maximal ideal.

3. C

   If $A \ne \mathbb{Q}$, then A has infinitely many prime ideals.

4. D

   For any nonzero a, b ∈ A, a divides b or b divides a in A.

Single choice · Rings and Fields

1. A

   The ideal $2\mathbb{Z}[i]$ is maximal in $\mathbb{Z}[i]$.

2. B

   The ideal $X\mathbb{C}[X, Y]$ is maximal in $\mathbb{C}[X, Y]$.

3. C

   The set of all polynomials in $\mathbb{C}[X]$ whose coefficients add up to 0 is a maximal ideal in $\mathbb{C}[X]$.

4. D

   The ideal $(\sqrt{2} - 1)\mathbb{Z}[\sqrt{2}]$ is maximal in $\mathbb{Z}[\sqrt{2}]$.

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   If f is onto, then so is $f^{n}$ for all n ≥ 1.

2. B

   ker $f^{n+1}$ $= \ker f^{n}$ for some n ≥ 1.

3. C

   If f is onto, then f is one-to-one.

4. D

   If f is one-to-one, then f is onto.

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   f(X) is irreducible in $\mathbb{Q}[X]$, but g(X) is not.

2. B

   g(X) is irreducible in $\mathbb{Q}[X]$, but f(X) is not.

3. C

   Both f(X) and g(X) are irreducible in $\mathbb{Q}[X]$.

4. D

   Neither f(X) nor g(X) is irreducible in $\mathbb{Q}[X]$.

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   Let f(X) ∈ 𝔽_p[X] and $\alpha$ be a root of f in 𝔽̄_p. Then 𝔽$_p(\alpha)$ is the splitting field of f in 𝔽̄_p.

2. B

   Let f, g ∈ 𝔽_p[X] be irreducible polynomials of same degree and $\alpha$ be a root of f in 𝔽̄_p. Then 𝔽$_p(\alpha)$ is the splitting field of g in 𝔽̄_p.

3. C

   𝔽_p[X] has infinitely many irreducible polynomials.

4. D

   The set {a + b | a, b ∈ 𝔽_p} is contained in {$a^{2} + b^{2}$ | a, b ∈ 𝔽_p}.

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   $x^{2} +$ xy$^{2} - x - y + 1$ is irreducible in $\mathbb{Q}[x,y]$.

2. B

   $x^{3} + 100x^{2} + 25x + 10$ is irreducible in $\mathbb{Z}[x]$.

3. C

   $2x^{2} + 3$xy $+ y^{2} + 3x + 2y + 1$ is irreducible in $\mathbb{Q}[x,y]$.

4. D

   $x^{4} + 4x^{2} + 3$ is irreducible in $\mathbb{Z}[x]$.

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   R has a unique maximal ideal.

2. B

   Every finitely generated ideal of R is principal.

3. C

   R is an integral domain.

4. D

   R is a euclidean domain.

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   The transposition (1 2) belongs to G.

2. B

   |G| is divisible by p.

3. C

   A p-cycle belongs to G.

4. D

   $G = S_{p}$

Single choice · Rings and Fields

1. A

   $\psi$ is surjective.

2. B

   If $p_{1}(x)y + p_{2}(x) \in \ker \psi$ for polynomials $p_{1}(x), p_{2}(x) \in \mathbb{C}[x]$, then both $p_{1}$ and $p_{2}$ are the zero polynomial.

3. C

   There exist non-zero polynomials $p_{1}(x), p_{2}(x) \in \mathbb{C}[x]$ such that $\psi(p_{1}(x)y + p_{2}(x)) = 0$.

4. D

   There exists $f \in \mathbb{C}[x, y]$ such that $\psi(f(x, y)) = 0$ and $\psi(f^{2}(x, y)) \ne 0$.

Single choice · Rings and Fields

1. A

   I = (x, y)

2. B

   I = (x + y)

3. C

   $I = (x^{2} + y^{2})$

4. D

   I = (xy − 1)

Single choice · Rings and Fields

1. A

   R is not commutative

2. B

   The ideal (X − 1) is a maximal ideal in R

3. C

   The ideal (X − 1, 2) is a prime ideal in R

4. D

   The ideal (X, 5) is a maximal ideal in R

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   $\mathbb{Q}[x]/(x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1)$

2. B

   $\mathbb{Q}[x]/(x^{5} + x^{4} + x^{3} + x^{2} + x + 1)$

3. C

   $\mathbb{Z}[x]/(x - 101)$

4. D

   $\mathbb{Q}[x]/(x^{43} - 41x + 41)$

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   The ring of polynomials in 3 variables over $\mathbb{R}$.

2. B

   The ring of complex analytic functions on the open unit disc in $\mathbb{C}$.

3. C

   The ring of 2 × 2 real matrices.

4. D

   The ring of continuous functions from [0, 1] to $\mathbb{R}$.

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   If K is the splitting field of a non-constant polynomial over $\mathbb{Q}$, then K is Galois over $\mathbb{Q}$.

2. B

   If K is a normal extension of $\mathbb{Q}$, then K is Galois over $\mathbb{Q}$.

3. C

   If K is the set of all the roots of the polynomial $x^{121} - x$ in an algebraic closure of 𝔽$_{11}$, then K is Galois over 𝔽$_{11}$, where 𝔽$_{11}$ is the field with 11 elements.

4. D

   If $K = \mathbb{Q}(2^(1/13), \zeta_{13})$, where $\zeta_{13}$ is a primitive 13th root of unity in $\mathbb{C}$, then K is Galois over $\mathbb{Q}$.

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   R is never an integral domain.

2. B

   r = −r for all r ∈ R.

3. C

   Every nonzero prime ideal of R is maximal.

4. D

   R must be a commutative ring.

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   $\mathbb{R}[x]/(f)$ is a field.

2. B

   $\mathbb{R}[x]/(f)$ is a finite dimensional $\mathbb{R}-$vector space.

3. C

   $\mathbb{R}[x]/(f)$ is a direct sum of fields, each of which is isomorphic to $\mathbb{R}$ or $\mathbb{C}$.

4. D

   There are no non-zero elements $u \in \mathbb{R}[x]/(f)$ such that u^m = 0 for some m ≥ 1.

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   The value of the Euler $\varphi-$function is even for all integers n ≥ 3.

2. B

   Let G be a finite group and S a subset of G with |S| > |G|/2. Then {ab : a, b ∈ S} = G.

3. C

   The polynomial ring $\mathbb{R}[x_{1}, \dots, x_{n}]$ is a Euclidean domain for all integers n ≥ 1.

4. D

   The subset {f ∈ C([0, 1]) : f(1/2) = 0} of the ring C([0, 1]) of continuous functions from [0, 1] to $\mathbb{R}$ is a prime ideal.

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   S is a cyclic group.

2. B

   S has $\varphi(2025)$ elements, where $\varphi$ denotes the Euler $\varphi-$function.

3. C

   S has $\varphi(2025)$ generators, where $\varphi$ denotes the Euler $\varphi-$function.

4. D

   S has 81 elements.

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   If f(a) is a unit in S for every non-zero element a ∈ R, then S is a field

2. B

   If f(a) is a unit in S for every non-zero element a ∈ R, then f(R) is a field

3. C

   If R is a field, then f(a) is a unit in S for every non-zero element a ∈ R

4. D

   If a is a unit in R, then f(a) is a unit in S

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   1

2. B

   8

3. C

   16

4. D

   24

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   Every quotient ring of R is a principal ideal domain

2. B

   There exists a quotient ring S of R and an ideal I ⊆ S which is not principal

3. C

   R has countably many ideals

4. D

   Every quotient ring S(≠ {0}) of R has a unique maximal ideal which is principal

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   S is a principal ideal domain

2. B

   $S/(y^{2} + x^{2})$ is a unique factorization domain

3. C

   S is a unique factorization domain

4. D

   S/(x) is a principal ideal domain

Multi-select · full marks only for the exact set · Rings and Fields

1. A

   $q = 2^{18}$

2. B

   $q = 2^{32}$

3. C

   $q = 2^{12}$

4. D

   $q = 2^{243}$

Single choice · Rings and Fields

1. A

   0

2. B

   2

3. C

   4

4. D

   8

Single choice · Rings and Fields

1. A

   0

2. B

   1

3. C

   2

4. D

   3