All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ʻalayhi wa sallam, is His slave and Messenger.
Firstly, we commend your good ambition and keenness to acquire knowledge; we implore Allaah to facilitate your affairs and bless you with success by His grace and generosity.
There is no religious impediment to the Muslim woman's traveling abroad for the purpose of completing her studies provided that she shall be safe from temptations in that place and that one of her mahram men accompanies her in her journey back and forth. It has been reported on the authority of Ibn ʻAbbaas that the Prophet, sallallaahu ʻalayhi wa sallam, said, “No man enters upon a woman except in the presence of her mahram. No woman travels except in the company of a mahram.” A man said, “O Messenger of Allaah! I have been enrolled for such and such military expedition, and my wife wants to go for Hajj.” He, sallallaahu ʻalayhi wa sallam, said to him, “Go with your wife.” This hadeeth confirms the requirement of a mahram in traveling, because that man intended to go out for jihaad (fighting in the cause of Allaah) while his wife wanted to go for Hajj, so he was ordered to go with her.
It is not necessary that the woman's mahram should accompany her throughout her stay; rather, he should accompany her in her journey back and forth. It is obligatory, though, to ensure that she is staying in a place where she can be safe from harms and temptations.
Lastly, it is worth highlighting that the statements mentioned by the dear questioner as a hadeeth reading, “Seek knowledge from cradle to death bed,” and “Acquire knowledge even in China,” are not ahaadeeth that are attributable to the Prophet, sallallaahu ʻalayhi wa sallam, as advised by Muslim scholars. Al-ʻAljooni cited them in his book Kashf Al-Khafaa' wa Muzeel Al-Ilbaas ʻamma Ishtahar min Al-Ahaadeeth ʻala Alsinat An-Naas. Nevertheless, their meaning is correct.
Allaah knows best.