What is the ruling of Islamic law on burping loudly after eating or otherwise, and repeating "Alhamdulillah" audibly thereafter? And is this considered among Islamic manners?

Deliberately raising one's voice while burping is contrary to Islamic etiquette and noble manners. The Messenger of Allah, peace and blessings be upon him, forbade it when he said to a man who burped in his presence: "Restrain your burp from us." A Muslim should learn the manners of eating, sneezing, and yawning, and avoid any behavior that repels those present. As for praising Allah after burping, there is no harm in it, for praise is desirable in all circumstances.