Gemara

74b (Mishnah - R. Yehudah): If one divorced his wife because she made a vow, and Rabim knew about the vow, he may not remarry her. (She is fined for taking a vow that cannot be annulled.)

R. Meir says, he may remarry her only if he could have annulled it without a Chacham.

Question: What is R. Yehudah's source (that a vow b'Rabim cannot be annulled)?

Answer: "Bnei Yisrael did not kill them (the Giv'onim) because the leaders of the congregation had sworn to them (in public. Such an oath cannot be permitted.)

Gitin 46a: Chachamim hold that the oath did not take effect at all (the Giv'onim deceived them by saying that they came from far away. Yisrael did not kill them due to Kidush Hash-m.)

Question: What is considered b'Rabim?

Answer #1 (Rav Nachman bar Yitzchak): It is three. It says regarding (one who sees blood of Zivah) "many days." "Days" teaches two; "many" teaches a third.

Answer #2 (R. Yitzchak): It is 10. It says "Edah" (like it says about the 10 evil Meraglim).

Gitin 34b (Mishnah): A widow collects her Kesuvah from orphans only by swearing (that she did not yet receive it). Judges refrained from administering the oath. R. Gamliel enacted that she vow.

35b (Rav Huna): This is only if she did not yet remarry. If she already remarried, she does not vow (lest she vow falsely, relying on her husband to annul it).

(Rav Nachman): She collects her Kesuvah through a vow even after remarrying.

Question: Surely, (she is not afraid, for) her husband will annul the vow!

Answer: She vows b'Rabim. (He cannot annul this.)

Question (against Rav Huna - Beraisa): If she remarried, she vows and collects her Kesuvah.

Answer: Tana'im argue about whether or not a husband can annul a vow b'Rabim. (This Tana holds that he cannot, therefore, she can still vow.)

36a (Mishnah): If a Kohen married a woman forbidden to Kohanim, he may not serve in the Mikdash until he vows not to benefit from her.

(Beraisa): Right after vowing, he may serve. After serving, he divorces her.

Question: Rav Nachman holds that a Chacham can permit a vow without hearing the details. Perhaps the Kohen will permit his vow and keep his wife!

Answer #1: He must vow b'Rabim.

Question: According to the opinion that a vow b'Rabim can be annulled, how can we answer?

Answer #2: He must vow "Al Da'as Rabim" (on the volition of others);

(Ameimar): The Halachah is, even according to the opinion that a vow b'Rabim can be annulled, a vow Al Da'as Rabim cannot be.

Rishonim

Rif (Gitin 18b): If she already remarried, she does not vow, for her husband would annul it. The Halachah is, even according to the opinion that a vow b'Rabim can be annulled, a vow Al Da'as Rabim cannot be.

Rambam (Hilchos Shevu'os 6:8): A vow in private is the same as a vow b'Rabim. If he regrets it, he may ask and we permit it.

Kesef Mishneh: The Rambam rules that b'Rabim cannot be annulled for the reasons of the Rosh (below). Tosfos, the Rashba, the Ran and the Mordechai agree.

Rosh (Gitin 4:11): Ameimar taught that even according to the opinion that a vow b'Rabim can be annulled, a vow Al Da'as Rabim cannot be. In Makos (16a), most texts say 'the Halachah is, a vow b'Rabim can be permitted. A vow Al Da'as Rabim cannot be permitted.' Even according to our text, Ameimar did not say merely 'even according to the opinion...' He prefaced 'the Halachah is...' to teach the Halachah. If a Neder b'Rabim could not be annulled, he would not need to teach that Al Da'as Rabim cannot be annulled. The question here was a monetary matter, in which the Halachah follows Rav Nachman, who says that we make her vow b'Rabim (because it cannot be annulled). However, we asked afterwards '... according to the opinion that it can be annulled, how can we answer?' We ask according to the Tana who says so, and also according to the Amora'im who argue with Rav Nachman. Rav Acha imposed a Neder Al Da'as Rabim on a Rebbi who hit children. It seems that a Neder b'Rabim did not suffice, for he holds that it can be annulled! It is difficult to say that there were not Rabim present. Rav Yosef said 'I will vow b'Rabim...' (Pesachim 107a; he holds that it cannot be annulled). The Halachah does not follow him.

Note: Perhaps Rav Acha did not impose a Neder b'Rabim lest they not find a suitable replacement, and he holds that a Neder b'Rabim cannot be permitted even with their Da'as! Perhaps the Rosh rejects this, for Rav Acha did not reinstate him; Ravina did. Or, perhaps all agree that also a Neder b'Rabim can be permitted for the sake of a Mitzvah. If this is not true, there is a challenge to the Rosh's Kal va'Chomer (if b'Rabim cannot be annulled, we need not say that Al Da'as Rabim cannot be annulled)!

Ran (DH Omar): R. Yehudah and Chachamim argue about this (Gitin 46a). The Halachah follows Chachamim, who say that b'Rabim can be permitted.

Question (Gilyon Efrayim): Chachamim hold like R. Meir in the Mishnah! (The Halachah normally follows R. Yehudah against R. Meir.)

Poskim

Shulchan Aruch (YD 228:23): A Neder b'Rabim can be annulled since it was not said 'on their Da'as.'

Taz (39) and Shach (74): He vowed on his own volition in front of Rabim.

Beis Yosef (DH v'Neder): It seems that the Rambam agrees with the Mefarshim that a Neder b'Rabim can be permitted, for he discusses only a Neder Al Da'as Rabim. (Bedek ha'Bayis - he writes that a Neder vowed to a Rabim can be permitted. I found a corrected text that says 'b'Rabim can be permitted'.) The Rashba (3:428) brings several sources in Gitin that it need not be permitted in front of the Rabim who heard the vow, and that they need not be told. One must be told only if the vow was contingent on his Da'as.

Shach (75): This is when the Rabim have no gain from the Neder. Even when an individual gains from a Neder, it may be permitted only in front of him, due to suspicion (Sa'if 20)!

Etzei Levonah: In Sa'if 20, one opinion, i.e. the Rambam (Shevu'os 6:7, according to the Tur), always requires permitting in front of those who heard the vow, due to suspicion. When an individual gains from a Neder, it may be permitted only with his consent! The Shach discusses the Rashba, who holds that there is suspicion only when the one who heard the vow gained from it.