The next question is why women in particular should cover their hair.
Someone is bound to bring up "tznius". Such a wonderful, all encompassing, term which can justify any restriction in dress, demeanour, speech or action. Its only bounds are the whims of the rabbi and the wrath of the Faithful in enforcing it. There is a touch of lip-service given to men being modest, but there are many fewer, and they don't tend to increase without end the way they do on women.
Women have to cover all their hair. By preference they should remove it completely. Women must cover everything but their faces and hands. There can't be even a hint of their figures in evidence. Women must not be heard singing. In short, everything that identifies women as women must be covered up and completely hidden.
On the other hand, men are required to show their hair and to grow extra bits that stick our from under their hats. They are all supposed to grow big jutting secondary sexual characteristics (beards). Women are not forbidden from hearing them sing. Some of the stricter sects say that men must wear form-fitting stockings that accent the male calf.
What it comes down to is absolutely classic. Women are inherently evil and seductive. Their sexual powers distract men and drag away from higher, holy things. So women have to be shorn of all signs of their basic female appeal. Couple it with a terrible fear that if women step out of line they will immediately screw every man in sight and - gasp - slip a woodscolt into the family. None of this is reciprocal, of course. Men are not forbidden from appearing male on the grounds that it will incite uncontrollable lust in women. Strictly speaking, adultery is only a crime for a married woman, not a married man.
I'm not saying that covering hair is wrong. My wife wears a headscarf - much more practical, attractive and versatile than a snood or baseball cap and doesn't require hacking off one of her prettiest features.
We just shouldn't be under any illusions as to why it's such a big issue. It's not about preventing adultery. People make the same sort of mistakes about sex all over the world. And that roughly 10% of all babies born in wedlock who were not sired by the husband is about the same Jew or Gentile and has been since we first developed scientific paternity tests. It's about control, particularly about the control of women's lives.