Gemara

In the great Beis ha'Keneses of Alexandria, when it was time to answer Amen, the overseer would wave flags, and everyone would answer (many people were too far away to be able to hear).

Berachos 47a - Beraisa: One may not answer an orphaned Amen;

Ben Azai says, if one says an orphaned Amen, his children will be orphaned.

Rishonim

Rif and Rosh (Berachos 34b and 7:17): One may not answer an orphaned Amen. If one does so, his children will be orphaned. The Yerushalmi says that an orphaned Amen is when one is obligated to say a Berachah, and he answers Amen without knowing to what he answers.

Question: In Alexandria people answered without hearing the Berachos!

Answers #1 and #2 (R. Yonah DH Ein and Rosh, ibid.): Those people already prayed and were not obligated to say the Berachos. Alternatively, it was permitted because they knew which Berachah they were answering to, for they are said in order.

R. Yonah (ibid.): According to this, if one knows which Berachah was said, he can be Yotzei by answering even if he did not hear the Berachah.

Answer #3 (R. Nisim, cited by Aruch, brought in Tosfos 52a DH v'Kivan): If one is obligated to say a Berachah, he may not answer without hearing it. In Alexandria they answered to Kri'as ha'Torah.

Rambam (Hilchos Berachos 1:13): If one heard a Yisrael say any Berachah, even if he did not hear the entire Berachah, and even if he is not obligated to say the Berachah, he must answer Amen.

Rambam (14): If one is obligated to say a Berachah and he did not hear it (but knows that it was said(, he may not answer Amen along with everyone else.

Kesef Mishneh: The Rambam permits one who need not say the Berachah to answer Amen even if he did not hear it. He learns from the Beis ha'Keneses of Alexandria (like R. Yonah).

Question (Lechem Mishneh): If he did not hear the Berachah at all, we already know (from Halachah 1:13) that one may answer Amen only if he heard the end of the Berachah, even if he is not obligated! We could say that Halachah 13 discusses when one is obligated to answer, and Halachah 14 discusses when one may answer. But why must the Rambam teach that he was not Yotzei? Even if he heard part of the Berachah he was not Yotzei! If he heard the end of the Berachah, we already learned this in Halachah 13!

Answer (Lechem Mishneh): Really, he heard the end of the Berachah. One might have thought that if many others answer Amen, surely they do not answer an orphaned Amen, and he may answer with them. The Rambam teaches that this is not so. This is unlike Rav Hai Gaon.

Poskim

Shulchan Aruch (OC 124:8): One should not answer an orphaned Amen. This is when Reuven is obligated to say a Berachah, and the Shali'ach Tzibur says it, and Reuven answers without hearing it, even though he knows which Berachah was said.

Mishnah Berurah (31): Even though it is a Mitzvah for everyone to hear the repetition of Shemoneh Esre, the Mechaber permits to answer Amen if one is not obligated to say the Berachah.

Rema: Some are stringent and forbid to answer Amen even if he is not obligated in the Berachah, for this is also an orphaned Amen, unless he knows which Berachah was said.

Beis Yosef (DH u'Mashma and DH Lachen): The Aruch also cites Rav Kohen Tzedek, who says that the Tzibur must be silent and hear every Berachah and then answer Amen. To answer without hearing is an orphaned Amen. The Tur explains that Rav Kohen argues with R. Nisim, and forbids answering to any Berachah without knowing which Berachah it was.

Beis Yosef (DH Aval): Tosfos explains differently, that they do not argue. Rather, R. Nisim supports himself from Rav Kohen! It was enacted that everyone listen and answer to the repetition of Shemoneh Esre, even those who know how to pray themselves. Therefore, we must say that the flags were waved during Kri'as ha'Torah, for those Berachos are not obligatory for the Tzibur. Tosfos himself gives an alternative answer, that it is forbidden only when one does not know to which Berachah he answers.

Taz (4): The argument about an orphaned Amen applies only to Shemoneh Esre. No one is stringent about other Berachos that a person says. R. Yonah teaches an orphaned Amen applies only to one who must say the Berachos. Nevertheless, in any case l'Chatchilah one must listen to the Berachos. It is Asur to answer without hearing if one could have heard. In the great Beis ha'Keneses, it was impossible for most people to hear, so it was permitted to answer upon seeing the flags. R. Yonah holds that knowing which Berachah was said does not help. The Rema: is stringent for the opinion of R. Yonah, and also that of for Rav Kohen, who says that an orphaned Amen applies even if one does not need to say the Berachah.

Question: The Yerushalmi says 'an orphaned Amen is when one is obligated in the Berachah and does not know to what he answers'. How can Rav Kohen oppose the Yerushalmi? The Bavli does not argue!

Answer (Taz 4): Rav Kohen explains the Yerushalmi to say 'One is obligated to hear the Berachah or to know to what he answers. If he does not know to what he answers, this is an orphaned Amen.'

Mishnah Berurah (32): Some are stringent to say that because it was enacted to repeat Shemoneh Esre, everyone is obligated in it, therefore even one who prayed may not answer without hearing the Berachah. L'Chatchilah one should be careful to hear the Berachos, but b'Di'eved one may answer as long as he knows which Berachah was said.

Kaf ha'Chayim (47): Some are stringent not to answer any Berachah he did not hear, even if he is not obligated to say it, and even if he knows which Berachah was said. It is best to follow this, for those who permit answering do not obligate answering, and the Isur of an orphaned Amen is severe.

Rema: One should not delay to answer. He should answer immediately after the Berachah is finished.

Mishnah Berurah (34): One must answer Toch Kedai Dibur, the time needed to say three or four words. After this is an orphaned Amen.

Mishnah Berurah (35): In Kaddish the end of the request to which we answer is 'uvi'Zman Kariv' or 'd'Amrin b'Olma'. Therefore, if the Shali'ach Tzibur sings 'v'Imru Amen' slowly, one should not wait for him to finish before answering Amen.

Kaf ha'Chayim (49): Hilchos Ketanos says that if the Shali'ach Tzibur waited and then began the next Berachah, answering now is like an orphaned Amen. If he did not wait at all, one may answer, for the Shali'ach Tzibur cannot deprive the Tzibur of the privilege to answer. Others say that in either case one may not answer.

Shulchan Aruch (11): If one finished Shemoneh Esre after the Shali'ach Tzibur finished a Berachah but before most of the Tzibur answered Amen, he answers Amen.

Rema (11): Even if he did not hear the Berachah, just he heard the Tzibur answering Amen and he knows to what they answer, he answers.

Source (Tziyunim l'Rema): This is from the Orchos Chayim citing Rav Hai Gaon (who permits only b'Tzibur, but not one who heard an individual answer Amen. It is brought in Beis Yosef DH Kasuv b'Orchos.)

Rebuttal (Magen Avraham 17): The Rema does not distinguish between the Tzibur and an individual. He discusses b'Tzibur merely because he comments on the Shulchan Aruch, who discusses b'Tzibur.

Rema: The same applies to Kaddish, Kedushah and Borchu.

Magen Avraham (18): Only something which an individual must say is considered obligatory. If not, how were people in Alexandria Yotzei Kaddish and Borchu?!

Question (Pri Megadim 56:5): In 56:1, the Rema allows to answer Kaddish if he hears the Tzibur answering, i.e. even if the majority already answered!

Answer (Bi'ur Halachah DH v'Kodem): R. Akiva Eiger says that one must finish Shemoneh Esre before the majority of the Tzibur answered, in order that the end of his prayer will not be considered an interruption between the Berachah and his Amen. The Magen Giborim says that a Shali'ach Tzibur waits only for the majority of the Tzibur to answer Amen.

Mishnah Berurah (56:9): If one finds the Tzibur at the end of 'Amen Yehei Shmei Raba...', he answers only 'Yehei Shmei Raba', for this is an independent praise, but not Amen. If he entered while the Tzibur is saying 'Amen Yehei Shmei Raba...', and he can intend for what they answer to, he also says Amen Yehei Shmei Raba.