14:34 Let your women keep silent in the churches, for they are not permitted to speak; but they are to be submissive, as the law also says. 35 And if they want to learn something, let them ask their own husbands at home; for it is shameful for women to speak in church.
36 Or did the word of God come originally from you? Or was it you only that it reached? 37 If anyone thinks himself to be a prophet or spiritual, let him acknowledge that the things which I write to you are the commandments of the Lord. 38 But if anyone is ignorant, let him be ignorant.
39 Therefore, brethren, desire earnestly to prophesy, and do not forbid to speak with tongues. 40 Let all things be done decently and in order.
Like the passage earlier on head coverings this passage has become a lightning rod. Research what is said about it and you will find a broad range of theories and conclusions, everything from women are never to speak a word in church for all time and in all places to the opposite position that Paul was addressing a specific time/place/culture and it has nothing to do with us. For those that hold closer to the first interpretation that teaching is a man's position, they can't be very good at it since they disagree with each other wildly.
Paul only speaks about this issue twice. The other time was in Timothy where he was addressing issues in Ephesus. Ephesus and Corinth are close together and shared the same culture, customs, and laws.
Before we travel to far abroad let's actually be faithful to the context of what Paul is addressing in the chapter in question which is order in the worship services rather than the chaos of competing voices that is currently taking place. More specifically in the immediate context he was referring to two or three at the most sharing a prophetic word one at a time and then those words being discussed and evaluated. It was in this context that Paul said the women should remain silent. He then refers to the law to back up this position. Note he never quotes the Bible. Paul always quotes the Bible when he is supporting a Biblical teaching. Why not here? In fact he just quoted the old testament a few verses earlier regarding speaking to people of other tongues. Could it be he is not referring to a Biblical law?
How could this issue of the silence of women be Biblical and Deborah be a judge of the nation in the time of the Judges? How could Huldah and Anna be prophetesses if women could not prophesy or comment on prophecies given? Why would women be counseled by Paul in this same letter to cover their heads while praying or prophesying if they were to always be silent? Why would Paul appoint Priscilla and Aquila to lead a church if women couldn't ever speak to the church?
People who hold to a greatly restricted role for women cling to passages like these ripped from their context and ignore much more direct and clear passages, themes, and evidence. Do you remember when five sisters came to Moses in the wilderness because their father had died? The law (civil, not Divine) said women could neither inherit nor own property. These women had no brothers meaning the land would be given to a male relative who already had land and they would have nothing. Moses could have just said "The law is the law" and walked away. Instead he asked God for direction. God told him to give the land to the sisters. Revolutionary from a cultural and civil law point of view but completely expected when one knows the character of God. In spite of this "shocking" revelation from God the cultural norms and civil laws of Israel did not change regarding the status of women. God revealed His will and humanity ignored it. Fifteen hundred years later the Roman Empire, considered to be quite advanced and enlightened still legally treated women as property to be bought and sold between fathers and husbands.
Now back to the context of 1 Corinthians 14. Paul is trying by every means available to him to rein in the chaos running wild in the church. One way was to have the women (who before becoming Christians never had a voice) hold their questions for discussion at home. Even that was a liberal teaching. Culturally women didn't need to ask questions anywhere. That was a man's business.
Now try and imagine a world where women had no voice coming into a relationship with Jesus and discovering that in Him there is no gender bias. There is neither make nor female. Suddenly a woman's world explodes. She has a mind, a voice, and lots of questions! Now put that new found freedom into a worship service where chaos already rules and everyone is either singing or prophesying or speaking in some unknown language.
Paul is just trying to dial back the craziness and one way to do that was to appeal to the law of the land and have the women hold their questions for later.
Think about this, if God wanted women to be silently about matters of faith why was the first messenger sent by Jesus to herald His resurrection a woman?