Are women required to cover their hair in Islam, where is the relevant verse(s) in the holy Quran?

Hijab is obligatory upon every adult Muslim woman. It must cover her entire body except for the face and hands; Hanafi scholars permit uncovering the feet. The evidence for the obligation of hijab in this manner is mentioned in the hadith narrated by 'A`isha (may Allah be pleased with her) in which she said that 'Asmaa` bint Abu Bakr entered upon the Messenger of Allah (pbuh) while wearing transparent garments.
 
He turned his face away and said, “O, Asmaa`! If a woman reaches menarche nothing should be seen [of her body] except this and this and he pointed to his face and hands" [reported by at-Tabarani).

Abu Dawud reported this hadith in the words, "O Asmaa`! If a woman has reached puberty nothing should be seen of her body except this and this' and he pointed to his face and hands.”

This meaning is mentioned by Allah in Surat An-Nur,

They should let their headscarves fall to cover their necklines. [An-Nur: 31)
Allah also says in Surat al-Ahzab,

O Prophet, tell your wives, your daughters, and women believers to make their outer garments hang low over them. That is more suitable that they will be known and not be abused. And ever is Allah Forgiving and Merciful. [Al-Ahzab: 59)

The ruling
Based on this, hijab in Islam is not considered one of the characteristics that distinguish a Muslim woman; rather, it is a means to fulfilling Allah's command to cover her 'awrah (parts of the body that must remain concealed)

The obligation to cover the 'awrah is not limited to women only; rather, it is a general command that encompasses males and females. The difference here is not in the nature of the ruling but in the extent of the 'awarah of a man and a woman. The extent of the 'awarah in a woman's body is larger than that of a man. This is due to her physical nature and modesty. According to the opinion of the majority of scholars, a woman's 'awrah before non-mahrams (persons to whom marriage is permissible) is her entire body except for the face and hands. Imam Abu Hanifa permitted women to uncover their feet as previously mentioned. This is because Allah Almighty prohibited women to show their adornment except that which is apparent, and feet are from among this; other than this, a woman must cover her entire body.