Are there any texts from the Prophetic Sunnah or incidents from the lives of the Companions indicating the permissibility of women going out to work? Is it true that 'Umar, may Allah be pleased with him, appointed a woman to head the marketplace during his caliphate? If so, in which source can this be found?

It is permissible for a woman to go out to work that suits her nature and physical constitution, provided that the Sharia-compliant controls are met. Indeed, the female Companions used to go out for jihad, seeking knowledge, and working in worldly affairs. Examples of this include Nusaybah bint Ka'b, Umm Sulaym, and Jabir's aunt. Furthermore, Umar ibn al-Khattab appointed Ash-Shifa bint Abdullah al-Adawiyah to oversee the market (al-Hisbah).