Gemara

Yevamos 87b (Gemara): If she married 'without permission' of Beis Din, i.e. based on witnesses, and her first husband returned, she may return to him.

Inference: The Reisha discusses with permission of Beis Din, i.e. through one witness.

92a (Mishnah): If she was married according to Beis Din (and her husband returned, she is exempt from a Korban).

(Rav Nachman): Beis Din's permission was a Hora'ah. (Therefore, she is exempt.)

(Rava): The permission was a mistake!

Support (Beraisa - R. Eliezer): She brings a fat Chatas.

117b (Mishnah): If one witness said that Reuven died, and his wife remarried, and one witness came and said that he did not die, she does not leave.

Inference: She does not leave because she already remarried. Had she not yet remarried, she would be forbidden to remarry.

Question: Ula taught that wherever the Torah believes one witness (e.g. the first, who says that Reuven died), he is like two witnesses!

Answer: The Mishnah means that if one witness said that he died, we permitted her, and one witness said that he did not die, she does not leave her Heter.

121b: A Nochri was saying, 'Who is from Chasa's house? Chasa drowned!'

Version #1 - Rav Nachman: Ha'Elokim (this is like an oath)! Fish ate Chasa!

Chasa's wife heard Rav Nachman say this and remarried. No one commented.

Inference (Rav Ashi): This shows that the Isur to remarry when a man fell into endless water is only l'Chatchilah. B'Di'eved, if she remarried she may remain.

Version #2: Rav Nachman permitted Chasa's wife to remarry. He reasoned that Chasa is a great man, so had he come to safety we would have heard.

This is wrong. Whether or not he is great, l'Chatchilah his wife may not remarry. B'Di'eved, she may stay married.

122b (Beraisa): We do not interrogate witnesses of Edus Ishah;

R. Akiva and R. Tarfon say, we interrogate them. (The Meforshim have this text. Our text attributes the first opinion to R. Akiva.)

Tana'im argue about R. Chanina's law:

(R. Chanina): Mid'Oraisa, both monetary and capital cases require interrogation - "You will have one law";

Chachamim exempted monetary cases from interrogation, lest people be afraid to lend money (lest the witnesses be disqualified).

The first Tana (in the Beraisa) considers Edus Ishah to be a monetary case, since she receives her Kesuvah (if we allow her to remarry);

R. Akiva and R. Tarfon hold, since we permit a married woman to remarry, it is like a capital case.

Rishonim

Rif and Rosh (43b and 15:9): If one witness said that he died, we permitted her to remarry, and one witness said that he did not die, she does not leave her Heter.

Rambam (Hilchos Gerushin 12:19): If one witness said that her husband died, Beis Din permitted her, and then a witness said that her husband is alive, she keeps her Heter to remarry.

Rambam (19): If the two witnesses came together, she may not remarry. If she did, she must leave, because it is a Safek. If she married the witness who permits and she says that she knows that her husband died, she does not leave.

Beis Shmuel (EH 17:123): When she married the permitting witness and she is sure, she may remain. This does not prove that if a woman married without a Heter from Beis Din (and no witness says that her husband is alive) she may remain. Here, both of them are sure that she is permitted.

Tosfos ha'Rosh (87b DH Alma): Even though one witness is believed (for Edus Ishah) mid'Oraisa, it is called 'with permission of Beis Din' because one witness is not believed about everything, and she must come to Beis Din to ask. When Beis Din accepted two witnesses and investigated, it is known that she is permitted and she does not need a Hora'ah of Beis Din.

Tosfos (87b DH Michlal): When we thought that one witness is believed (for Edus Ishah) mid'Oraisa, it is called 'with permission of Beis Din' because one witness is not believed about everything, and she must ask Beis Din.

Tosfos (117b DH Ha): If Beis Din permitted her through one witness, and then a witness said that her husband is alive, she may not remarry. 'She does not leave her Heter' means that b'Di'eved, if she married she need not leave.

Beis Shmuel (ibid.): We may not infer that if she remarried through one witness without permission of Beis Din that she must leave when the second witness comes. Perhaps permission allows her to remain even if she remarried after the second witness came, but even without permission she may remain if she remarried before the second witness came.

Nimukei Yosef (47a): A Beis Din of commoners suffices for Edus Ishah and all laws of Gitin and Kidushin. Chachamim enacted that commoners suffice for all common monetary cases in which a party might lose money, lest people be afraid to lend or buy. Here also, these matters are common and (inability to marry is) more serious than loss of money.

Poskim

Shulchan Aruch (39): A woman who remarries based on one witness needs permission of Beis Din. If she remarried without permission of Beis Din some say that she need not leave, even if a witness came and said that her husband did not die.

Beis Yosef (DH Kasuv bi'Teshuvos l'Ramban): The (Rashba, Teshuvah 119 attributed to the) Ramban and Rivash (brought below) say that when there is one witness a woman needs permission of Beis Din. Without this, the witness does not help. However, the Ran (Teshuvah 47) and Radach (8:2) say that if she remarried based on one witness without a Heter from Beis Din and then a witness testified that her husband is alive, she need not leave.

Rivash 508 (cited in Beis Yosef DH Kosav ha'Rivash): Even if a woman has enough testimony, she should not remarry without a Heter of Beis Din, since she has only one witness. This is even if Beis Din accepted the testimony. We learn from Yevamos 92a and the Tosfos ha'Rosh. Whenever Chachamim said that we may not testify, e.g. a body was found later and perhaps it changed, she must leave, and all the fines apply. Ha'Rit agrees.

Beis Yosef (DH uv'Mah): R. Yerucham (Nesiv 23:3:195:3) says that we excommunicate one who permits a woman improperly, even when the Halachah is that she need not leave. This is unlike the Rivash.

Chelkas Mechokek (77): All agree that she need not leave! The questioner in Teshuvas ha'Ran thought that since she remarried without a Heter from Beis Din, it is as if the witnesses came together and she must leave. However, the Ran said that since she acted upon the first witness, he is considered like two, so we ignore the second. The Ramban said that one witness does not help without permission of Beis Din. This is only if she did not yet remarry.

Source (Gra 130): We learn from 121b. A woman remarried based on what she heard Rav Nachman say. Even though Beis Din did not permit her, she was allowed to stay married.

Rema: The Beis Din must be three Kosher men, not related to each other or to the witnesses.

Rebuttal (Chelkas Mechokek 78): The Darchei Moshe (8) learned this from Teshuvos ha'Rashba (749 and 750). There is no proof from there. One Chacham suffices, like all Hora'os, just we require three to accept the testimony lest people say that it was fabricated.

Defense (Gra 131): Every matter that requires Beis Din requires three judges. Tosfos (87b DH Michlal) concludes that since one witness is not believed mid'Oraisa, she truly needs permission of Beis Din.

Beis Shmuel (124): The Chelkas Mechokek learns from Ed mi'Pi Ed, which is valid even though Beis Din did not hear the first witness. Normally, Beis Din must hear the testimony is so it will not be Ed mi'Pi Ed, and here Ed mi'Pi Ed is Kosher! However, the Gemara (Yevamos 122b) supports the Rema. It says that the argument about whether or not Edus Ishah needs Drishah v'Chakirah depends on whether it is like monetary cases or capital cases. All hold that it is a proper Din (judgment), unlike other Hora'os of Isur v'Heter. We do not learn from Ed mi'Pi Ed, for there it is impossible for Beis Din to hear the first witness. Even R. Akiva, who requires Drishah v'Chakirah, accepts Ed mi'Pi Ed, even though there is no Drishah v'Chakirah of the first witness. The Nimukei Yosef permits commoners on the Beis Din just like common monetary cases in which a party might lose money. I.e. also here commoners are Sheluchim of judges with Semichah. If we do not require Beis Din, surely commoners are fine! The Chelkas Mechokek also learns from a Pesak of the Terumas ha'Deshen. This is wrong. There, there was a Beis Din. Also, perhaps we more lenient if she already remarried.