GATE MA 2025 Paper 1 — page 20

Questions 58–60 of 65. Imported answers require review unless explicitly marked reviewed.

Let

be an uncountable set. Let the topology on

be defined by declaring a subset

to be open if

is either empty or finite or countable, and the empty set to be open. Then, which of the following is/are TRUE?

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

Every compact subset of

is closed

Every closed subset of

is compact

(singleton subsets are closed) but not

(Hausdorff)

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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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2025 · Paper 1

Question 58

Question 59

Question 60