This article also covers the issue:
Hair Covering for Married Women
By Rabbi David Sedley
Obligation
The Mishnah (Kesuvos chapter 7) states that a woman can be divorced and forfeits her Kesuba (divorce settlement) if she violates either Das Moshe (religion of Moshe) or Das Yehudis (religion of Yehudis – it is not clear whether Yehudis here refers to a specific woman or the conduct of Jewish (Yehudi) women in general).
Das Moshe refers to Torah commandments, and includes such things as if a wife feeds her husband or children non kosher food, not keeping the laws of family purity, or making vows and not keeping them. Das Yehudis refers to laws of modesty based on convention (and enforced with rabbinic rulings). The Mishnah includes such things going out with uncovered hair, spinning in the market place and flirting with men in this category.
The Talmud there (72a) challenges the Mishnah (which states that women’s hair covering is a Rabbinic law), and claims that the prohibition of women going outside with their hair uncovered is a Torah law. It is learned from the verse “The Cohen shall uncover the woman’s hair” (Bamidbar 5: 18). The Talmud resolves this apparent contradiction by stating that the Torah only requires a minimal, symbolic hair covering (she may wear a basket on her head), whereas the Rabbinic law requires her to cover her hair entirely.
This law is brought as normative Halachah by Rambam (Hilchos Issurei Biah 21: 17) and the Shulchan Aruch (Even HaEzer 21: 2): Jewish women should not go bareheaded in the marketplace.
Later (ibid. 115: 4) the Shulchan Aruch states: These are the acts that violate Das Yehudis: Going into the marketplace, an open thoroughfare or a crowded square bareheaded, without a scarf such as all other women wear, even if her hair is covered by a kalta (small scarf with holes in it).
There is discussion amongst the commentators as to whether this prohibition applies also to unmarried women or divorced or widowed women. The consensus seems to be that it does not apply to unmarried women, and there are lenient opinions as to divorced or widowed women (see for example Igros Moshe Even HaEzer vol. 1. 57 and vol. 4 32: 4).
Covering Hair inside the Home
Regarding the requirement for a woman to cover her hair inside the home, the Talmud (Yoma 47a) praises Kimchis, who merited to have seven sons who were all High Priests in the Temple. This was in the merit of never uncovering her hair even in her own home. All halachic authorities stress that it is correct and modest for a woman to cover her hair in her own home, but most concede that it is not a binding halachah (see for example Igros Moshe Even HaEzer vol. 1. 58).
Leaving Some Hair Uncovered
Similarly, although it is the correct thing for a woman to cover all of her hair, the consensus seems to be that technically it is permitted to show up to one tefach of hair (ibid. see also, Magen Avraham Orech Chaim 75: 2 who states that based on the Zohar it is forbidden for any of her hair to show, and that this is the appropriate custom).
In a Place Where Most Women Do Not Cover Their Hair
Nowadays, many women do not cover their hair, even outside the home. This has two possible implications on halachah.
According to some opinions, because many women do not cover their hair a man is not obligated to divorce his wife if she refuses to cover her hair, and she does not forfeit her Kesuba (Igros Moshe vol. 1. 114. But see Yabia Omer vol. 3, Even HaEzer 21 who is stringent about this).
There may also be another leniency. It is forbidden for a man to pray or recite a blessing in the presence of erva (a part of a woman’s body which is considered licentious or is normally covered). The Aruch Hashulchan (Orech Chaim 75: 8) says that even though it is forbidden for a woman to go outside without her hair covered, since many women do so, it is no longer considered erva regarding prayer, and it is permitted to pray and say blessings in the presence of a woman with her hair uncovered (although it is forbidden to look at her while praying or saying the blessing). However, the Mishna Brura (75: 10) strongly disagrees and forbids any blessing or prayers to be said in the presence of a married woman with her hair uncovered. This applies even to a husband in his own home if his wife has not fully covered her hair.