Gemara

(Mishnah): One can be a Bechor (firstborn) for inheritance (to receive a double portion), but not a Bechor for Pidyon ha'Ben.

The case is, he was born after a Nefel (non-viable baby), even if the Nefel stuck its head out alive, or after a viable baby (that was in the womb for nine months) that stuck its head out after it already died.

Inference: If the first baby was viable and stuck its head out alive (and later died), the next child would not be a Bechor even regarding inheritance (because the first child was considered born)!

Bechoros 46b (Shmuel): If the head of a Nefel left the womb, this is not considered birth (to exempt the next baby from Pidyon).

He learns from "Kol Asher Nishmas Ru'ach Chayim b'Apav" the law depends on the nose (or head) only for something that can live.

Question (Mishnah): A baby born after a Nefel whose head came out alive, or after a nine-month baby whose head came out dead (is a Bechor for inheritance, but not for Pidyon).

This shows that birth of a Nefel's head is considered birth regarding Pidyon!

Answer: The Mishnah means that the majority of the body left. The Tana said "head" for parallel structure with the Seifa, which discusses a viable baby whose head came out dead;

There, he needed to say "head", so we can infer that had the head came out alive, the next child would not be a Bechor for inheritance, and not for Pidyon.

Rejection: Another Mishnah teaches that Chidush!

(Mishnah): If a baby is born normally (head-first), once the majority of the head comes out, i.e. the forehead, it is considered born.

Shmuel is refuted.

Nidah 23b (Mishnah - R. Meir): If a woman's first birth had the form of an animal or bird, her next birth (if he is a boy) is a Bechor for inheritance.

(Rava): Even if it could live, inheritance depends on "Reishis Ono," the first child whose death would cause the father to be Mis'onen (mourn).

Rishonim

Rambam (Hilchos Nachalos 2:10): One born after a Nefel, even if the Nefel stuck its head out alive, is a Bechor for inheritance. Similarly, if a nine-month baby's head came out dead, the next child is a Bechor for inheritance. The Torah says Reishis Ono, i.e. that no previous child was born alive. Therefore, if a nine-month baby stuck most of its head out alive (and later died), the next baby is not a Bechor.

Rashi (68a DH Af): The Mishnah discusses twins. After the Nefel's head came out and went back in, the viable fetus left. The Nefel does not uproot the latter from being the Bechor for inheritance. Even if the Nefel were totally born, the next child is a Bechor for inheritance, for one does not grieve over death of a Nefel. All the more so, the next if a Bechor for inheritance if the Nefel's head came out dead. The Mishnah says that (only) the Nefel's head came out to teach that this is birth, and the next child is not a Bechor for Pidyon.

Poskim

Shulchan Aruch (CM 277:6): One born after a Nefel, even if the Nefel stuck its head out alive, is a Bechor for inheritance. Similarly, if a nine-month baby's head came out dead, the next child is a Bechor for inheritance The Torah says Reishis Ono, i.e. that no previous child was born alive.

Beis Yosef (DH Bechor): (Birth of) the majority of the head is like the head. Rashi says that the Mishnah discusses twins. After the Nefel's head came out and went back in, the viable fetus left. Even though birth of the head is considered birth, inheritance depends on Reishis Ono - one over whom the father would grieve if he died. This excludes a Nefel. (Therefore, the viable baby is Reishis Ono.)

SMA (13): Emergence of the forehead is considered birth, i.e. for a viable baby. A Nefel is not considered a child. Inheritance depends on Reishis Ono. Based on this, even if a Nefel was totally born, the next child is a Bechor for inheritance. This is unlike She'eris Yosef (25), who says that if a Nefel was totally born, the next is not a Bechor for inheritance.

Drishah (5): Shmuel was rejected. It seems that the Mishnah discussed when only the Nefel's head came out, for if the entire Nefel came out, the next child would not be a Bechor for inheritance. Rashi in Bechoros connotes like this. This is why he needed to establish the Mishnah to discuss twins (so we can say that the viable baby was born before the Nefel totally left). The Tur connotes like this. He says that if a woman miscarried the form of an animal, the next child is a Bechor for inheritance. This connotes a child born after a Nefel is not a Bechor for inheritance. She'eris Yosef (25) says so.

Rebuttal (Drishah DH Aval): This is wrong. Also when it is fully born, its father would grieve it says death. Shmuel was rejected, but we cannot learn from this to inheritance. Shmuel taught about Pidyon, that the Nefel uproots the latter from Pidyon only if the majority came out, but not if only the head came out. He was rejected. Even emergence of the head exempts the latter from Pidyon. No one distinguishes the head from the majority regarding inheritance. Rashi said that the Nefel's head returned because the Mishnah says that the next child is not a Bechor for Pidyon. I.e. even birth of the head uproots the latter from Pidyon. Also the Mishnah of a miscarriage of an animal form teaches a Chidush regarding Pidyon, but the same applies (also regarding inheritance) to a full birth of a standard Nefel.

Note: Hagahos Tur ha'Shalem (17') says that the Nimukei Yosef (Bava Basra 66a) connotes like She'eris Yosef. I did not see such a connotation. In DH Pesikna, the Nimukei Yosef says that emergence of the majority of the head helps only for a viable baby. However, there he does not discuss Bechorah, rather, one who gave a gift to his first child that will be born. In DH Garsinan, he says that only a one day old affects the calculation of the extra share of the Bechor, for it says "v'Yaldu Lo."

Shach (1): Also the Bach (5) says like the SMA. Rashi explicitly says so in Chulin.

Taz, citing She'eris Yosef: Do not say that the Mishnah teaches the head for the Chidush that even though only the head left, the next child is not a Bechor for Pidyon. The Rambam and Tur bring this, and they discuss only inheritance! Therefore, if a baby was born and lived a few hours, and it is possible that it is an eight-month baby, the next baby is not a Bechor for inheritance.

Taz: The SMA is correct. However, She'eris Yosef ruled correctly in his case. The Mishnah discusses a Vadai Nefel. If it is a Safek Nefel, the next child is a Safek Bechor, so he does not receive double. Also, we follow the majority of babies, which are viable. Aveilus is an exception; we are lenient about Safek Aveilus.

Sidrei Taharah (YD 194:27 DH Yotzei): If the first child was a Yotzei Dofen (born through Caesarian section), and the next baby was born normally, it is not a Bechor for Pidyon (YD 305:24). The Taz (20) says that this is because it is not a full Bechor, i.e. for inheritance, since it is not Reishis Ono. This implies that if the first were a Nefel, the latter would be a Bechor also for Pidyon. This is wrong, since the latter is not the Bechor child. Regarding animals, there is only Bechor for children, and we say (19a) that if the first was Yotzei Dofen, the second is not a Bechor!

Chasam Sofer (EH 2:161 DH u'Mah): I know of no source to say that inheritance depends on viability. The Mishnah says that if a Nefel's head came out alive, the next child is a Bechor for inheritance. She'eris Yosef says that if it was totally born alive, the next child is not a Bechor for inheritance, for the Nefel is the Bechor. The SMA and Shach disagreed, but in another Teshuvah I brought good support for She'eris Yosef. Cases occurred, and I persuaded fathers to compromise with their Safek Bechor (about how much he will inherit). Perhaps the Torah mentioned Reishis Ono regarding Bechor to teach that regarding standard inheritance, a Nefel inherits like a regular child.

Pischei Teshuvah (6): Sha'ar Mishpat says that this refers to a Vadai Nefel. If it is a Safek, even if the baby died at birth, and all the more so if he died later within 30 days, we follow the majority of babies, which are viable. This is why mid'Oraisa, a baby exempts his mother from Chalitzah and Yibum even if he died after birth (EH 156:4). Surely, the next baby is a Safek Bechor, and ha'Motzi mi'Chavero Alav ha'Re'ayah. He is not even a Safek Bechor, for he was not recognized to be a (Vadai) Bechor when he was born, like the Rema says in Sa'if 11.

Sefas Emes (Bechoros 46b DH bi'Gemara): Surely, Shmuel knew the Mishnah! Rather, Shmuel holds like Reish Lakish, that the forehead is not enough to be considered birth regarding inheritance. We do not learn this from the other Mishnah. The Gemara holds like R. Yochanan, therefore it says that Shmuel is refuted.

Shulchan Aruch (ibid.): Therefore, if a nine-month baby stuck most of its head out alive, the next baby is not a Bechor.

R. Akiva Eiger: Ben Lev (3:44) was unsure if this is only if we know that it was a nine-month baby, or if we assume that a Stam baby is a nine-month baby. She'eris Yosef considered a Stam baby that died after a few hours to be a Nefel. I say that most babies are viable. If a woman married a Kohen (and her son died within his first month) we do not require Chalitzah. Even if we consider the baby a Safek Nefel, the next is a Safek Bechor, who does not get a double portion.