GATE MA Algebra PYQs & Answers

Questions 1–10 of 54. Imported answers require independent review unless marked reviewed.

Let

be a group of order 595. Which one of the following is TRUE?

Your answerChoose one option.

cannot have a proper normal subgroup.

must have a proper normal subgroup.

cannot have an element of order 17.

The number of Sylow 5-subgroups is 17.

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Let

be the ring of integers modulo 11. Let

be the ring of all polynomials with coefficients in

. Which of the following statements is/are TRUE?

Your answerSelect all correct options. You can choose more than one.

The polynomial

is irreducible over

is a field.

The ideal

is maximal.

The ideal

is not a prime ideal.

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Let

, where

is the ring of all polynomials with coefficients in

. Then the degree of the field extension

is equal to ________. (answer in integer)

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Which one of the following statements is TRUE?

Your answerChoose one option.

The set of all

real matrices under usual matrix addition and multiplication is a Unique Factorization Domain.

The set of all real valued functions under usual function addition and multiplication is a Unique Factorization Domain.

The set of all polynomials with coefficients in

under usual polynomial addition and multiplication is a Unique Factorization Domain.

The set of Riemann integrable functions in

under usual function addition and multiplication is a Unique Factorization Domain.

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Let

be the ring of all polynomials with coefficients in

under the usual polynomial addition and multiplication. Let

be defined by

Which of the following statements is/are TRUE?

Your answerSelect all correct options. You can choose more than one.

is a ring homomorphism.

is one-to-one.

is onto.

is a ring isomorphism.

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Let

be the group of integers modulo

. Let

be the external direct product of

and

. Then which of the following statements is/are TRUE?

Your answerSelect all correct options. You can choose more than one.

is not a cyclic group.

The order of the element

is 98.

The order of the element

is 60.

There is an element in

of order 360.

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Let

be an algebraically closed field containing a finite field

. Let

be the subfield of

consisting of elements of

that are algebraic over

Consider the following statements:

S1:

is algebraically closed.

S2:

is infinite.

Then, which one of the following is correct?

Your answerChoose one option.

S1 is TRUE and S2 is FALSE

S2 is TRUE and S1 is FALSE

both S1 and S2 are TRUE

neither S1 nor S2 is TRUE

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All rings considered below are assumed to be associative and commute with

. Further, all ring homomorphisms map

. Consider the following statements about such a ring

P1:

is isomorphic to the product of two rings

and

P2:

such that

and

P3:

ideals

with

such that

and

P4:

with

such that

Then, which of the following is/are TRUE?

Your answerSelect all correct options. You can choose more than one.

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Let

and

be field extensions which are not algebraic. Let

be algebraic over

and

. Let

be the subfield of

generated over

by the coefficients of the monic polynomial of minimal degree over

which has

as a zero. Then, which of the following is/are TRUE?

Your answerSelect all correct options. You can choose more than one.

is a finite extension if and only if

is a finite extension

The dimension of

over

is greater than the dimension of

over

The dimension of

over

is smaller than the dimension of

over

is an algebraic extension if and only if

is an algebraic extension

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Let

be a group with identity element

, and let

be such that the following hold:

(i)

(ii)

, and

Then, the least positive integer

for which

is _____ (in integer)

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GATE MA Algebra PYQs

How to practice Algebra

Question 14

Question 23

Question 29

Question 41

Question 52

Question 54

Question 47

Question 59

Question 60

Question 64