To what extent is the prophetic hadith, cited in Imam Al-Qurtubi's exegesis of Surat Al-Alaq, authentic: "Do not allow your women to dwell in rooms, and do not teach them writing"?

Teaching reading, writing, and beneficial sciences is a legitimate endeavor for both men and women. Seeking knowledge is an obligation upon every Muslim. The texts of the Quran and Sunnah (definition) encourage the pursuit of beneficial knowledge that teaches an individual about their religion and their worldly life, purifies their soul, and improves their character. One should not abandon learning due to the bad manners of some learners. The hadith (definition) mentioned by Al-Qurtubi: "Do not house them in upper chambers, nor teach them writing" is a fabricated hadith. Teaching women increases their intellect, and their deficiency in religion and intellect does not mean they should abandon learning. Indeed, the Prophet, peace and blessings be upon him, commanded Al-Shifa bint Abdullah to teach Hafsa, the Mother of the Believers, ruqyah (definition) and writing.