Charliehall, some covering is de’Oraisa as the Gemora (Kesuvos 72A) says. Full covering — de’Rabbonon.
Inrejection, the Gemora on the same daf says that a wife who does not cover her hair “leaves (the marriage) without her kesuvah (paid)”. There is a machlokes in Sotah (I can double-check which daf) if it is a chiyuv to divorce her or a mitzvah. The thing about her not covering the hair at the beginning is a Rav Moshe’s chidduch. Chacham Ovadia differs.