What is the history of Islam's entry into India, when did that happen, and what is the authenticity of the hadith narrated in "Mustadrak Al-Hakim" regarding the King of India gifting the Prophet, peace and blessings be upon him, pickled ginger?

Islam entered India through the Islamic conquests led by Muslim leaders. In the year 44 AH, Al-Muhallab ibn Abi Sufra invaded India. In the year 48 AH, Ziyad ibn Ubayd directed Sinan ibn Salamah al-Hudhali to invade India. Then, Muhammad ibn al-Qasim conquered the city of Debal and other areas in the year 93 AH. In the year 95 AH, Muhammad ibn al-Qasim conquered the city of Multan. The Abbasid Caliph Al-Mahdi equipped Abd al-Malik ibn Shihab al-Misma'i to invade India in the year 160 AH. In the year 392 AH, Sultan Mahmud ibn Sebuktigin invaded India and conquered many lands. He used to invade India every year and broke the idol of Somnath.