Can zakat be given to a Student of knowledge? 

In the Quran, Allah tells us to whom Zakat should be given.  Allah says 

إِنَّمَا ٱلصَّدَقَـٰتُ لِلْفُقَرَآءِ وَٱلْمَسَـٰكِينِ وَٱلْعَـٰمِلِينَ عَلَيْهَا وَٱلْمُؤَلَّفَةِ قُلُوبُهُمْ وَفِى ٱلرِّقَابِ وَٱلْغَـٰرِمِينَ وَفِى سَبِيلِ ٱللَّهِ وَٱبْنِ ٱلسَّبِيلِ ۖ فَرِيضَةًۭ مِّنَ ٱللَّهِ ۗ وَٱللَّهُ عَلِيمٌ حَكِيمٌۭ

"As-Sadaqat (i.e. Zakah) are only for the poor; and those in need; and those employed to collect it; and to attract the hearts of those who have been inclined (towards Islam); and to free the captives; and for those in debt; and for Allah's cause; and for wayfarers; a duty imposed by Allah. And Allah is All-Knower, All-Wise." [al-Tawbah 9:60]

Among the categories mentioned in this ayah is: فِى سَبِيلِ ٱللَّه  "For Allah’s cause." Scholars have differed over the meaning of “for Allah’s cause” in this ayah. Some scholars have restricted it to only include those fighting legitimate jihad on the battle field. However, other scholars have included the seekers of knowledge. This is because seeking knowledge is a type of jihad.  

Shaykh Uthaymeen said, "The scholars said, ‘for Allah’s Cause’ also includes a man who devotes all his time to seeking Islamic knowledge, so he should be given from the zakah funds enough to cover his expenses for clothing, food and drink, accommodation and the books of knowledge that he needs, because Islamic knowledge is a kind of jihad for the sake of Allah."

The proof that seeking knowledge is a type of jihad can be found in the Quran, Sunnah and statements of the Salaf. For example, Allah says   

فَلَا تُطِعِ ٱلْكَـٰفِرِينَ وَجَـٰهِدْهُم بِهِۦ جِهَادًۭا كَبِيرًۭا

"So obey not the disbelievers, but make Jihad against them with it (i.e. the Quran) the great Jihad," 25:52 
So memorizing the Quran and teaching it are from the greatest types of Jihad. This proves that jihad is not restricted only to those fighting on the battle field. 

Likewise, from the Sunnah we have the authentic hadeeth (Al-Albaani), collected by Imam Abu Dawood from Anas who said the Prophet صلى الله عليه وسلم said, 

جَاهِدُوا الْمُشْرِكِينَ بِأَمْوَالِكُمْ وَأَنْفُسِكُمْ وَأَلْسِنَتِكُمْ

"Make Jihad against the polytheists with your wealth, lives and tongues," In other words, teaching Islamic knowledge is a means of doing Jihad, for this reason the seeker of knowledge is considered a mujahid.  
Also from the statements of the Salaf, is the statement of Muadh Ibn Jabal who said, "Acquire knowledge because doing so is good, Seeking it is ibaadah, Studying it is glorifying Allah, searching for it is jihad . . ."  And the statement of  Abu Dardaa who said, "Whoever holds that going out in the morning or going out later in the day to gain some knowledge, is not jihad, then he is deficient in his intellect and deficient in his opinion." So seeking knowledge is a type of jihad, and for this reason scholars have ruled that students of knowledge are eligible to receive Zakat under the category, "for Allah’s cause."
Imam As-Sa’di said in his Tafseer of the Ayah from Surat At-Tawbah, "If a person is able to earn a living but he devotes all of his time to seeking knowledge, he may be given zakah, because seeking knowledge comes under the heading of jihad in Allah’s cause."
And Allah knows best.