It is always in your best interest to get the remaining portion of the tooth removed as it may cause a infection any time. Right now the antibiotics have cleared the infection but that piece will always keep acting as a source of infection.

Kindly get the broken part of the tooth removed as it will be causing infection often.

It might pain later. Permanent solution is to remove that broken tooth.

Yes, it's necessary! pain has gone due to antibiotic. It may recur again. Better to get the tooth extracted n get over with it!

Kindly consult a dentist in person for further suggestion. We need more investigations to decide upon treatment. You may need surgical removal procedures. Advance procedure can be done with laser too. You may consult me in person too. Dental tips: - visit a dentist every six months. Gargle your mouth thoroughly after every meal. Scrub gently to clean your tongue with a tongue cleaner. Floss all your teeth inter dentally & brush twice daily, morning & night, up & down short vertical strokes, with ultra-soft bristles, indicator brush.