Does a married woman have to cover her hair based on a biblical commandment or a rabinnic decree?
It depends on who you ask, and how they interpret the sources.
The Mishna on Ketubot 72a list it as Das Yehudit (which the gemara contrasts with DeOraysa):
Yet the gemara there presents another Tannaitic source which (as per the gemara's interpretation) regards it as Biblical:
The gemara then harmonizes it by explaining that there are two levels of requirement and how different levels of covering, in different locations, satisfy the different levels. See inside.
That is the normative halacha. (Though see how various Rishonim and Acharonim might interpret that gemara, perhaps moving it from this fixed point.)
What follows is my own chiddush in the matter. I would point out that one might disagree with the gemara and interpret that "warning" in Tana deVei Rabbi Yishmael as something other than a Biblical requirement.
And that if one looks in the Sifrei, a close but not identical picture emerges:
One might argue that this is evidence of the true intent of the derasha, even as it appears in Tana deVei Rabbi Yishmael, that this is revealing the Dat Yehudis, the tznius practice of women even back then, rather than instituting a Biblical requirement.
The gemara in Ketubot derives the requirement from the laws of the Sotah in bamidbar 5:18. As the woman is required to UNcover her hair, there must have been covering on it.
The discussion is much more complex, but you can read about it here.