It surprises people to learn that women living under an Islamic order could be scholars, that is, hold the authority that attaches to being knowledgeable about what Islam commands, and therefore sought after and deferred to.
The typical Western view is that no social order has (or aspires to have) more 'religion' in it than Islam. The more "religion' a society has in it, the more restricted the scope in that society for women to enjoy agency and authority. Behind that is the assumption that religion is 'really' a human construct, done mainly by men and therefore done to secure advantages for them at the expense of women. Muslims, of course, do not share this view.
In the Quran and Sunnah, Muslims believe they have a framework of guidance that is strictly impartial and sufficient because God's knowledge and mercy encompass all beings and all their pasts and futures. Any human derivation from and within that framework is subject to revision, but the framework itself is not. Accordingly, in the Islamic tradition, to say 'God says in His Book' decides the argument.
Where it is not certain how the guidance of the Quran is to be acted upon, Muslims look to the example of how God's Messenger acted in the same or a similar situation. The record of his example (Sunnah) is now, for all practical purposes, conveyed through a body of texts, known singly and collectively as hadith (lit. 'saying). A man who becomes an expert in knowledge of the hadith is called a muhaddith; a woman, muhaddithah (plural, muhaddithat).
The Quran rebukes the people of the jāhiliyyah (the Ignorance before Islam) for their negative attitude to women. When news is brought to one of them of [the birth of] a girl, his face darkens, and he is chafing within! He hides himself from his folk because of the evil he has had news of. Shall he keep it in disdain or bury it in the dust? Ah - how evil the judgment they come to! (Quran, 16:58-59)
The costly prospect of bringing up a daughter (a son was expected to enhance a clan's military and economic potential) perhaps explains this negative response to the birth of a girl. Burying infant girls alive was a custom among some (not all) of the Arab tribes of that time. The Quran warns of retribution for this gross atrocity on the day When the infant buried alive shall be asked for what sin she was killed (Quran, 81:8-9)
Human rights and duties indicated in the Quran are pegged to two fundamentals that are the same for men and women - namely they being creatures and slaves of God, their Creator, and Lord, and they being the issue of a single human self. God has said in the Quran; O humankind, be wary of your Lord who created you from a single self, and from it created its pair, and from the pair of them scattered many men and women. Be wary of God, through Whom you ask of one another [your rights and needs] and close kindred:! God is ever-watchful over you. (al-Nisa 4:1) And (al-Araf, 7:189): He, it is Who created you from a single self, and made from it its mate, so that he might settle at rest with her. Male and female are created for the same purpose: I have not created jinn and humankind except so that they worship Me (al-Dhariyat, 51:56). The Quranic term 'abd signifies both 'worshipper and 'slave' in relation to God.
The duties owed to God, and the virtues that ensue from the effort to do them, are the same for men and women. This is affirmed in a well-known Quranic verse. The verse and the occasion of its revelation are recorded in this hadīth, narrated by Abd al-Rahmān ibn Shaybah:
I heard Umm Salamah, the wife of the Prophet ﷺ say: I asked the Prophet ﷺ Why are we [women] not mentioned in the Quran as the men are mentioned? [...] Then, I was alerted that day by his call on the pulpit. [...] At that moment, I was combing my hair. I gathered up my hair and went to one of the rooms; I listened hard. I heard him saying on the pulpit: O people, God says in His Book: The Muslim men and Muslim women; the believing men and believing women; the men who are obedient [to God] and women who are obedient [to God]; the men who are truthful and the women who are truthful; the men who are persevering and patient and the women who are persevering and patient; the men who give alms and the women who give alms; the men who are humble and the women who are humble; the men who fast and the women who fast; the men who guard their chastity and the women who guard their chastity, and the men who remember God much and the women who remember God much - God has prepared for them forgiveness and a great reward. (AL-HAKIM, al-Mustadrak, ii. 416. The verse cited is al-Ahzab, 33:35)
Having 'the knowledge' and the conscientious preserving, transmitting and understanding it is the strong basis for the public authority that learned Muslims, men and women, were able to command. Sometimes there were different opinions on the import of the knowledge people had. Still, the differences were not settled based on the gender or the tribe, or socio-economic class of the person who conveyed it.
A striking case is that of Amrah bint Abd al-Rahmān, the great tabi'iyyah (Successor), muhaddithah and faqīhah, who intervened in a court case in Madinah to prevent a miscarriage of justice. It is remarkable enough that she knew that the case was in progress and the circumstances of it and what sentence the qādī had passed but not yet carried out. Many famous men jurists were residents and active in the city; none of them intervened. What is astonishing is that she did intervene, and no one questioned her right to do so. The defendant was a non-Muslim, not known to Amrah except as the defendant in this suit, in which she had no personal, private interest. The qãdi reversed his decision and released the defendant only because he could have no argument against the authority of the hadith she was able to cite. He did not know or remember it, or simply failed to bring it to bear when reaching his judgment. Once he knew the hadith, he did as a Muslim should – he acted upon it.
The Quran speaks about women in general and specific terms. It does not associate womanhood with inferiority or deficiency of any sort, any primordial sin, or any disposition to sin not also found in men, or any disposition to induce sin in others not also found in men. It does not regard women as an appendage of men but as distinct beings, each called individually, just as are men. The language of the Quran, Arabic, like many others, uses masculine forms to mean women also unless the context expressly excludes them. The grammar does not require women to be expressly included; therefore, when that explicit inclusion occurs, it is all the more striking. The above verse 33:35 enumerates the virtues, distinctly for men and women. Starting with the next verse in that sūrah, here are a few more examples:
It is not for a believing man or believing woman when God and His Messenger have decided a matter [...] (al-Ahzab, 33:36).
Never will I allow to be lost the work of any of you, male or female (Al Imran, 3:195).
Whoever does righteous deeds, male or female, and is a believer, him We shall enliven to a good life, and We shall pay them certainly a reward proportioned to the best of what they used to do (al-Nahl, 16:97).
Whoever does righteous deeds, from among the male or the female, and he is a believer, those will enter Paradise [...] (al-Nisa, 4:124).
The believing men and believing women are protecting friends (awliyā') of one another, they bid to good (al-maruf), and forbid from evil (al-munkar); they establish the prayer and give the alms (zakah) and obey God and His Messenger (al-Tawbah, 9:71).
The Quran and Sunnah are replete with examples that give women the right to attain high rank in all spheres of knowledge.