Gemara

103b (Mishnah): One may not cook any kind of meat with milk, except for meat of fish and grasshoppers;

One may not bring cheese on the same table with any kind of meat, except for meat of fish and grasshoppers.

104a (Rav Yosef): Our Mishnah holds that the Torah forbids cooking fowl with milk. This is like Chachamim, who argue with R. Akiva, who forbids fowl with milk only mid'Rabanan.

Rebuttal (Rav Ashi): It is like R. Akiva. One may not cook any kind of meat with milk. Some kinds are forbidden mid'Oraisa, and others, mid'Rabanan;

(Rav Yosef): The Mishnah forbids putting fowl on the same table with milk. This shows that the Torah forbids fowl cooked with milk. If it were only a Rabbinic decree, we would not forbid putting them on the same table. We do not make a decree to guard a decree!

(Mishnah): A non-Kohen may eat at the same table with a Kohen eating Chalah of Chutz la'Aretz (which is only mid'Rabanan). Such Chalah may be given to any Kohen (even an ignoramus);

Rebuttal (Abaye): In Chutz la'Aretz, there is no concern for any Torah Isur (regarding Chalah). There is no need to decree!

Regarding meat and milk, if we allow fowl and milk on the same table, one may come to put beef and milk on the same table, and eat them together!

Rebuttal (Abaye): If we allow fowl and milk on the same table, one may come to put beef and milk on the same table, and eat them together!

104b (Agra, the father-in-law of R. Aba - Beraisa): Fowl and cheese may be eaten wantonly.

(Agra): This means that one need not wash his hands or clean his mouth in between.

113a (Mishnah): Basar v'Chalav of Tahor Behemos may not be cooked together. If they were cooked, one may not benefit from it. If the meat or milk is from a Tamei Behemah, they may be cooked together, and one may benefit from it;

R. Akiva says, the Torah does not forbid meat of a Chayah or bird (with milk). It says three times "do not cook a kid in its mother's milk," to exclude Chayos, birds and Tamei Behemos.

R. Yosi ha'Gelili says, it says "do not eat any Nevelah," and "do not cook a kid in its mother's milk." The Isur of Basar v'Chalav applies to species to which Nevelah applies.

Suggestion: Since Nevelah applies to birds, Basar v'Chalav should apply to birds!

Rejection: "In its mother's milk" excludes birds, which do not have milk.

Rishonim

Rambam (Hilchos Ma'achalos Asuros 9:3): The Torah forbids only meat of a Tahor Behemah with milk of a Tahor Behemah. One may cook and benefit from meat of a Tahor Behemah with milk of a Tamei Behemah or vice-versa. One who eats it is not liable for Basar v'Chalav.

Rambam (4): The Torah does not forbid meat of a Chayah or bird in milk of a Chayah or Behemah. Therefore, one may cook and benefit from them. It is forbidden to eat mid'Rabanan, lest people transgress and come to an Isur Torah of Basar v'Chalav, and eat meat of a Tahor Behemah with milk of a Tahor Behemah. The Torah connotes only a kid in its actual mother's milk, therefore Chachamim forbade all Basar v'Chalav.

Magid Mishneh: Rav Ashi explained the Mishnah to say 'one may not cook any kind of meat with milk. Some kinds are forbidden mid'Oraisa, and others, mid'Rabanan.' The Rambam explains that 'cook' means eating. Rav Ashi adopted the wording of the Torah (which discusses only cooking meat and milk).

Rosh (8:5): The Rif brought Agra's Beraisa that permits eating fowl and cheese wantonly. This shows that he holds that milk and fowl is mid'Rabanan, like R. Akiva. This is like Rav Ashi, who said that our Mishnah is like R. Akiva. Also Abaye rejected Rav Yosef's proof that the Mishnah is like Chachamim. The Ge'onim rule like this.

Rosh (51): The Rif (42b) brought (the first Tana of the Mishnah (113a) and) R. Akiva, but not R. Yosi ha'Glili. This shows that he rules like R. Akiva. If he held that Chachamim argue with R. Akiva, he would rule like them, for we do not follow R. Akiva against a majority! Rather, he holds that the first Tana does not argue. The first Tana teaches that Basar v'Chalav applies only to Tahor species, and R. Akiva teaches the source to exclude Chayos, birds and Tamei species.

Tosfos (113a DH Basar): Rabanan forbid fowl mid'Oraisa. Only R. Akiva forbids fowl and Chayos mid'Rabanan. We learn like it says below, that one may not cook with milk what is forbidden due to Neveilah (i.e. species to which Neveilah applies).

Poskim

Shulchan Aruch (YD 87:1): Three verses say "Lo Sevashel Gedi ba'Chalev Imo." They forbid cooking, eating, and benefit.

Shulchan Aruch (3): The Isur applies only to meat of a Tahor Behemah with milk of a Tahor Behemah. One may cook and benefit from meat of a Tahor Behemah with milk of a Tamei Behemah or vice-versa, or meat of a Chayah or bird in milk even of a Tahor animal. Mid'Rabanan one may not eat it.

Beis Yosef (DH u'Gdi): The Rif, Rambam, Rosh, Rashba and Tur rule like R. Akiva. The Tur's words are difficult. He says that mid'Rabanan, one may not eat milk and fowl and meat of a Tahor Behemah with milk of a Tamei species. The Torah forbids the latter!

Bach (1 DH Isur): The first Tana, R. Akiva and R. Yosi ha'Glili all agree that there is an Isur mid'Rabanan of meat and milk even if one of them is Tamei species. Therefore, we say Chatichah Atzmah Na'asis Nevelah. (The Basar v'Chalav is considered like pure Isur, e.g. regarding Bitul in 60.)

Shach (3): No Posek says that Chachamim decreed to forbid meat and milk when one of them is a Tamei species. Rashi and the Ran explicitly say unlike this. The Beis Yosef means that since in any case no one will eat it, there is no reason to decree.

Bach (2 DH u'Mah): The Tur says that there is no Isur to cook or benefit from milk and fowl. This is unlike the Sugya! Rav Ashi answered that some meat and milk is Asur mid'Oraisa, some (i.e. milk and fowl) is Asur mid'Rabanan, and some (fish and milk) is totally permitted. The Magid Mishneh said that 'to cook' means eating. Rav Ashi adopted the wording of the Torah. This is difficult, for the Torah refers to cooking and eating! The Rambam (3) says that the Torah permits cooking and benefit from meat and milk when one of them is Tamei. Also regarding fowl, he says that the Torah does not forbid eating, therefore cooking and benefit are permitted, i.e. mid'Oraisa. He says that mid'Rabanan one may not eat them; the same applies to cooking and benefit. He did not explicitly say so, because the Gemara did not explicitly say so. Tosfos says that Chachamim forbid milk and fowl mid'Oraisa; this connotes that the Halachah follows them. We do not follow R. Akiva against a majority. The Rosh said that the Rif holds like R. Akiva, but it is difficult to be lenient against Tosfos. Also the Terumas ha'Deshen says so in Hagahos Sha'arei Dura. Fowl is just like beef regarding eating, cooking and Hana'ah, unlike the Rema.

Shach (2): I answered the Bach's question elsewhere. The Rambam and many others permit Hana'ah.

Taz (2): The Yam Shel Shlomo said that a word is missing from the Tur. The Tur merely teaches that one may cook and benefit from meat and milk if one of them is from a Tamei species.

Shach (4): I support those who say that the Isur is mid'Rabanan. Even the Terumas ha'Deshen said only that it is hard to be lenient against Tosfos and permit eating cheese right after fowl, but letter of the law he agrees that milk and fowl is mid'Rabanan. This is why he permits finished eggs with milk! The Rambam (Hilchos Mamrim 2:9) says that one who says that milk and fowl is mid'Oraisa transgresses Bal Tosif (adding to Mitzvos).

Toras Chatas (85:13): Isur v'Heter ha'Aruch (37:5:15) holds that if milk fell into fowl and some spilled (so we do not know if there was 60), it is forbidden, for Ta'am k'Ikar is mid'Oraisa.

Rebuttal #1 (Taz (98:5): Even if Ta'am k'Ikar is mid'Oraisa, milk and fowl is only mid'Rabanan! Isur v'Heter's words are according to the opinion that milk and fowl is mid'Oraisa. Even so, we can defend the Rema's ruling, for we find that Chachamim strengthened their decree of milk and fowl like Torah law, e.g. regarding eating fowl after cheese, or putting them on the table together. It is not considered a decree for a decree.

Rebuttal #2 (Shach 98:7): The Rashba, Ran and other Poskim say that since milk found in a slaughtered animal is (forbidden with meat only) mid'Rabanan, we are lenient when we do not know whether there is a Shi'ur for Bitul. Below (111:1,4) we are similarly lenient about a Safek whether was the Shi'ur Chachamim required for Bitul. Really, Isur v'Heter said that we are stringent like R. Tam, who says that Ta'am k'Ikar is mid'Oraisa, and even if milk fell into fowl, this is no different. He means that it is no different because he holds that milk and fowl is mid'Oraisa! Since we hold that it is mid'Rabanan, we are lenient.