How to understand if one is eligible for research?

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What are the eligibility criteria for one to undergo research in Mathematics?

Or should I place the question as what virtues of a student are given importance when one is interviewed for a PhD programme in some reputable institute?

Or may be how will a student know himself whether he is eligible for research?

As I am interested in research, I wonder what are the conditions one should keep in mind while preparing for an interview/an exam for undergoing the same.

The fact is there is no specific syllabus in this case which makes life more difficult.There is no one to guide you here on what will be the questions or how to answer them nor any books available

I know there are many in this site who are doing research or have successfully completed it.

Can I get some help from them?