Gemara

Descendants of Haman taught Torah in Bnei Berak.

Sotah 35b (Beraisa - R. Shimon): In Gilgal, Yisrael wrote the Torah on rocks.) At the bottom they wrote "(you will kill the seven Kena'ani nations) lest they not teach you to do like all their abominations."This shows that if Kena'anim outside of Eretz Yisrael wanted to convert, we would have accepted them.

(Beraisa): "And you will capture a captive (and desire her)" includes Kena'anim in Chutz la'Aretz. We accept them to be converts if they repent.

This is like R. Shimon.

Avodah Zarah 10b (Rebbi): "V'Lo Yihyeh Sarid l'Veis Esav" refers to those who act like Esav.

Mechilta (Beshalach 2 DH va'Yomer): R. Elazar ha'Moda'i says, Hash-m swore by His Kisei ha'Kavod that He will not leave a descendant of Amalek in the world, lest one say 'this camel is from Amalek.' R. Eliezer says, Hash-m swore that they will accept any convert, but not from Amalek or his house. When the Ger told David 'I am an Amaleki (and I killed Sha'ul, for he was about to die and told him to do so)', David remembered the Halachah, and said 'you will be killed, for you incriminated youself.'

Yalkut Shimoni (Shmuel 2:141): R. Yitzchak says, the Ger Amaleki was Do'eg ha'Edomi. David said "Damecha (plural)" are on your head. He killed Do'eg for killing the Kohanim of Nov. Some say that he was Do'eg's son.

Rishonim

Rambam (Hilchos Melachim 5:5): It is a Mitzvas Aseh to destroy the seed of Amalek.

Hagahos Maimoniyos (1): This Mitzvah will not apply until the days of Mashi'ach, after conquering Eretz Yisrael - "when Hash-m will give you rest from all your enemies around, in the land... Timcheh Es Zecher Amalek" (Sefer ha'Mitzvos Lav 236).

Rebuttal (Radvaz): The Parshah of Shmuel and Sha'ul refutes this.

Rambam (6:4): If we made war with the seven nations or Amalek, and they did not make Shalom, we do not leave anyone alive. It says about the seven "Lo Sechayeh Kol Neshamah", and it says about Amalek "Timcheh Es Zecher Amalek." "The only city that made Shalom with Yisrael was the Chivi in Givon." I.e. we offered to make Shalom with them, and they did not accept.

Rebuttal (Ra'avad): This is wrong. However, we can say that 'they made Shalom', i.e. they accepted the Mitzvos.

Kesef Mishneh: The Ra'avad holds that even if the seven nations and Amalek accept taxes and servitude, we kill them. The Rambam can hold that 'making Shalom' includes accepting the seven Mitzvos, for then they are not the seven nations or Amalek. They become Kosher Bnei Noach.

Tosfos (Sotah 35b DH Lerabos): Why did Bnei Yisrael accept Rachav to be a convert? She was from the seven nations, in Eretz Yisrael, and she converted and married Yehoshua (Megilah 14b)! Our Gemara connotes that all agree that we do not accept the seven nations in Eretz Yisrael. Perhaps Hash-m commanded to accept her (for a Hora'as Sha'ah, i.e. a temporary enactment for a great need). Hash-m said (Maharsha - do not say) that we accept them before the war started, for we wrote the Torah on the rocks before the war started, and we accepted Kena'ani converts only from Chutz la'Aretz! Perhaps we would have accepted even the nations in Eretz Yisrael had they repented at the time. "In order that they will not teach you to do" teaches that if they repent, we do not kill them (Sifri Shoftim). It does not say that we do not accept them. This refers to the nations we were commanded to kill. "Do not leave anyone alive" applies only after Yehoshua began the war. Yehoshua sent letters of Shalom to the nations before the war began. We must say that Rachav accepted to convert before the war began. R. Yehudah and R. Shimon argue only about accepting Kena'anim who were in Chutz la'Aretz after the war began.

Chazon Ish (YD 157:5): The Ra'avad holds that we do not offer Shalom to Amalek, and we do not demand taxes and servitude from them, but if an Amaleki accepted his seven Mitzvos, we may not kill him. It seems that Amalek can say 'make Shalom with me on condition that that I accept the seven Mitzvos.' Presumably this is only before making war. Once we began war with them, we do not accept them. They are no better than the seven nations, whom we do not accept after they began war. Descendants of Haman taught Torah in Bnei Berak. Even though Agag entered war, we accept his descendants not at the time of war. However, this is not a proof. Perhaps his descendants were mixed with others (and we did not know that they are Amalekim). In any case the Rambam and Ra'avad hold that if they come to convert, we accept them. The Mechilta says that we do not accept from Amalek, i.e. after the war began. One may not leave alive a descendant of Amalek is when they act like Amalek, like we say about Esav (Avodah Zarah 10b). David killed the Amaleki because he converted during the war.

Bo'i Chayei (CM 1:236, citing Meforshim): The 'Amaleki' was a Yisrael. He was called an Amaleki because he lived in their region.

Avnei Nezer (OC 508): The Rambam connotes that if they accept the seven Mitzvos, taxes and servitude, we accept them. Rashi holds that the seven nations in Eretz Yisrael cannot make Shalom, and Tosfos allows them to make Shalom only before beginning war. Even if the Rambam allow even after the war began, what is his source for Amalek?! We needed a verse for the seven nations, and there is no verse for Amalek. We cannot learn from the seven nations, for we kill them lest we learn from them. We kill Amalek due to what they did after Yetzi'as Mitzrayim! It seems that the seven nations were Chayav Misah for their abominations. It is a Chidush that Teshuvah helps. It does not help for Misas Beis Din. Amalek is punished for their ancestors' sin. Normally, the Torah does not do so. Hash-m knows that the hatred is in their hearts. The episode with Haman proves this. Hash-m punishes children for their fathers' sins only when they continue in them. If they accepted the seven Mitzvos, this shows that they repented. (Hagahah - "sons will not die for fathers" applies to Yisrael. It disqualifies testimony of relatives, but relatives can testify for Nochrim! A convert is not judged for his past. However, one who accepts only the seven Mitzvos is still a Ben Noach. The Ramah (59a) connotes that a Ben Noach is not punished for his father's sins. We must say that the Rambam agrees. This requires investigation.) The Mechilta holds that he is killed even after conversion (and surely after accepting only seven Mitzvos). We can answer according to Tana d'Vei Eliyahu (24). Elifaz commanded his son Amalek 'the world to come is only for Yisrael. Hep them, and you will merit with them.' Amalek became a hater of Yisrael. We can say that even after he converts, he does not accept to be submissive to Yisrael. If he will see a need to be submissive, he will hate and seek to kill them like his ancestor. However, when we conquer the land and they accept taxes and servitude, this shows that they accept to be submissive to Yisrael, and ceased their ancestors' actions.

Rambam (Hilchos Sanhedrin 18:6): Beis Din does not kill through one's admission, only through witnesses. It was a Hora'as Sha'ah for David to kill the Amaleki.

Rambam (Hilchos Isurei Bi'ah 12:17): Any Nochri who converts is immediately a Yisrael. He may marry within Yisrael immediately, except for four nations - Amon, Mo'av, Mitzrayim and Edom.

Ein Zocher (3:1, cited in in Frankel Rambam): The Rambam here and in Hilchos Sanhedrin connote that we accept converts from Amalek.

Sefer ha'Chinuch (Sof 604): It is a Mitzvah on the entire Tzibur, and every individual male, in every place and every time, if they can, to kill and destroy anyone from the seed of Amalek. If one had the oppositely to kill an Amaleki, and he did not, he was Mevatel this Mitzvah.

Minchas Chinuch: This Mitzvah does not apply nowadays, for Sancheriv relocated the nations. For any Nochri that we see, we assume that he separated from the majority (one of the other nations).

Acharonim

Question (Iyun Yakov Sanhedrin 96b): The Mechilta contradicts our Gemara. This requires investigation.

Answer (Maharsham 3:272): Perhaps these descendants were from Haman's daughter. Regarding Nochrim, lineage follows the father.

Yehoshu'os Malko: We learn from Haman's descendants that if we accepted conversion of an Amaleki, b'Di'eved it is valid.

Rebuttal (Seridei Esh 2:73:1): R. Yehudah and R. Shimon argue about accepting Kena'anim from Chutz la'Aretz. It seems that in Eretz Yisrael, their conversion is invalid (see Tosfos). If so, the same applies to Amalek.

Ya'avetz (Sanhedrin 78a): The Gemara exempts one who killed a Tereifah. If so, why did David kill the Amaleki? Perhaps killing a king is different. In any case, it was a Hora'as Sha'ah, for there were no witnesses or warning.

Beis She'arim (OC 18, Ha'aros): Initially, David thought that we accept converts from Amalek, for the Torah does not forbid this. "Do not abhor an Edomi" - Amalek is from Edom! The Chiyuv to kill Amalek is if they don't convert. Then, David remembered that Moshe was told that we do not accept them.

Avnei Nezer (OC 138:4): We learn from David that we do not accept a convert from Amalek. All the more so we do not accept him to be a Ger Toshav.

Meshech Chachmah (Sof Ki Tetzei): Rashi explains that the Mitzvah is not even to leave an animal of Amalek,like the Mechilta. We must say that the Mitzvah does not apply to individuals, for "David and his men... killed the Geshuri, Gizri and Amaleki... and took cattle and flock" (Shmuel 1:27:8,9). This shows that an individual need not kill animals. David killed the Ger Amaleki because a Nochri is killed through his own admission (Sefer ha'Chinuch 26,192).