Hello,
A P.hd I. in Inorganic Chemistry is a very good career. It mainly deals with the Synthesis, structure and behaviour of inorganic or organometallic compounds. Most research and product development positions in inorganic chemistry field requires a Doctorate in Inorganic Chemistry.

Almost every sector of chemical industry deals with inorganic chemistry.
They can hold positions like

• Mining Analytical Chemist
• Environmental Inorganic Chemist
• Ceramic Chemist/Engineer
• Food Safety and Inspection Services
• Academic Careers

Here's a list of universities in India offering PhD in Inorganic Chemistry https://university.careers360.com/colleges/list-of-phd-in-chemistry-universities-in-india
Hope it helps