GATE MA Algebra PYQs & Answers — Page 2

Questions 11–20 of 54. Imported answers require independent review unless marked reviewed.

Topic practice 54 questionsPage 2 of 6

Let

be the subring of

generated by

and

. Consider the following statements.

I. The ring

is a unique factorization domain.

II. The ring

is a principal ideal domain.

Which one of the following is correct?

Your answerChoose one option.

Both I and II are TRUE

I is TRUE and II is FALSE

I is FALSE and II is TRUE

Both I and II are FALSE

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Given a prime number

, let

denote the number of

–Sylow subgroups of a finite group

. Which one of the following is TRUE for every group

of order 2024?

Your answerChoose one option.

and

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The number of non-isomorphic finite groups with exactly 3 conjugacy classes is equal to ______ (answer in integer)

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Let

be the field with exactly 3 elements. The number of elements in

is equal to ______ (answer in integer)

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The number of group homomorphisms from

is equal to ______ (answer in integer)

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Consider the following statements.

I. There exists a proper subgroup

such that

is a finite group.

II. There exists a subgroup

such that

is isomorphic to

Which one of the following is correct?

Your answerChoose one option.

Both I and II are TRUE

I is TRUE and II is FALSE

I is FALSE and II is TRUE

Both I and II are FALSE

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Let

be the field extension of

obtained by adding all the roots of the polynomial equation

. The number of distinct fields

such that

is equal to

(answer in integer)

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Let

denote the group of

invertible complex matrices with usual matrix multiplication. For

denotes the subgroup generated by

and

. Let

and

be three subgroups of

given by

Let

denote the center of

for

. Which of the following statements is correct?

Your answerChoose one option.

is isomorphic to

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Let

, where

denotes the set of rational numbers and

denotes the set of integers. For

, let

denote the ideal generated by

Which of the following statements is/are correct?

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

is an irreducible element in

, then

is a prime ideal in

is a unique factorization domain

is a prime ideal in

is NOT a principal ideal domain

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Consider the rings

where

denotes the ideal generated by

and

denotes the ideal generated by

Which of the following statements is/are correct?

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

Every prime ideal of

is a maximal ideal

has exactly one maximal ideal

Every element of

is either nilpotent or a unit

There exists an element in

which is NEITHER nilpotent NOR a unit

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GATE MA Algebra PYQs

Question 15

Question 16

Question 27

Question 29

Question 31

Question 40

Question 59

Question 36

Question 45

Question 46