Gemara

(Beraisa): "Lo Yishma Al Picha" - do not vow or swear in the name of idolatry, or cause others to do so.

This supports Shmuel's father.

(Shmuel's father): One may not make a partnership with a Nochri, lest the Nochri need to swear, and he will swear in the name of idolatry, and the Yisrael transgresses "Lo Yishma Al Picha."

Avodah Zarah 6b (Mishnah): One may not collect a loan from Nochrim (three days before their festival. This is a decree, less one repay. Alternatively, the Nochri will rejoice that he no longer owes, and thank his idolatry.)

(Beraisa - R. Yehoshua ben Korchah): One may not collect a loan for which he has a document. He may collect a Milveh Al Peh (a loan without a document), for this is saving one's money (lest the Nochri deny it later).

(Rav Huna): The Halachah follows R. Yehoshua.

22a: Some Yisraelim and Nochrim were joint sharecroppers. The Nochrim worked on Shabbos, and the Yisraelim the next day. Rava permitted it.

Megilah 28a: Avuha bar Ihi said that his merit was that he never became a partner with a Nochri.

Rishonim

The Rif (Avodah Zarah 7a) brings Shmuel's father's teaching.

Rambam (Hilchos Avodah Zarah 5:10): One may not make a Nochri swear by his idolatry.

Rambam (Hilchos Shutafim 5:10): One may not become a partner with an idolater, lest the idolater become obligated to swear to him, and he will make him swear in the name of his idolatry.

Hagahos Maimoniyos (5): The Re'em explains that we forbid a partnership only when they pool their money and buy jointly. One may buy and sell with Nochrim. We forbid only before and after their festivals, and we permit going to their fairs.

Rosh (7:3): The Rashbam says that if one became a partner with a Nochri and the Nochri was obligated to swear, he should not accept it (rather, he should exempt him from swearing). R. Tam says that if he is already a partner, he may accept the oath rather than lose, just like one may collect a Milveh Al Peh, even though the Nochri wil thank his idolatry. However, one could distinguish. There it is a Safek (lest he thank his idolatry), and here he will surely swear in the name of his idolatry. R. Tam brought another proof from Avuha bar Ihi (Megilah 28a), who said that his merit was that he never became a partner with a Nochri. What is the Chidush? It seems that others were not so careful about Shmuel's father's law, but he was. I say that one may accept the oath. Shmuel's father did not say that a partner with a Nochri may not make him swear. Rather, he forbade becoming a partner, lest the Nochri swear, but b'Di'eved, he may accept it rather than lose his money. L'Chatchilah one must be careful not to become his partner. Since b'Di'eved one may accept the oath, most people were not careful about this. Nowadays it is permitted, for they swear in the name of their Kedoshim, whom they do not consider to be gods. Even though they mention Hash-m's name and intend for something else, they do not mention idolatry, and they intend for the Creator of Heaven and earth. Even though they join His name with something else, we do not find an Isur to cause this. Lifnei Iver does not apply, for they are not commanded about this.

Bedek ha'Bayis (YD 147) brings this from R. Yerucham in the name of the Ri. He says that nowadays they swear in the name of their Kedoshim called Even Gilyon. They say Hash-m's name with intent for their Messi'ah.

Ran (Avodah Zarah 7a DH Garsinan): The proof from collecting a loan is invalid. There, perhaps he will not thank his idolatry, but here he will surely swear! Also, one who collects surely saves his money, but imposing an oath is not sure to save! We cam bring a proof from Avuha bar Ihi. His merit was that he never became a partner with a Nochri, even if would exempt him from swearing, lest he often be near him and learn from his deeds. Alternatively, even at a time when Nochrim do not normally swear in the name of their idolatry, he did not become a partner with them. People rely on this nowadays to be lenient. The Ramban says that there is no Isur to become a partner with a Nochri. Lifnei Iver, even mid'Rabanan, applies only to what he could not do without the Yisrael, e.g. giving wine to a Nazir. Also, perhaps Nochrim are not forbidden to vow or swear in the name of idolatry. "Lo Yishma Al Picha" refers to Yisrael. It is not forbidden even mid'Rabanan because the Yisrael does not tell him to swear in the name of idolatry. He swears (in its name) by himself when he is obligated to swear. Avuha bar Ihi said that this was his merit, for it is only Midas Chasidus.

Hagahos Ashri (Avodah Zarah 2:1): R. Tam agrees that l'Chatchilah one may not join with a Nochri. However, Rava argues with Shmuel's father. Perhaps the Halachah follows Rava. However, the Rif brought Shmuel's father. This shows that he rules like him.

Beis Yosef (CM 182 DH u'Perush): He refers to the joint sharecroppers. Rava did not forbid tem. This is no proof that Rava argues with Shmuel's father. What could he do? To end the partnership would be a loss. R. Tam permits accepting an oath rather than lose. Even the Rashbam would not require breaking up, just he may not accept an oath.

Poskim

Shulchan Aruch (OC 156:1): One must be careful not to become a partner with Nochrim, lest the Nochri was obligated to swear to him, and the Yisrael will transgress "Lo Yishma Al Picha."

Rema: Some are lenient about partnership with Nochrim nowadays, for they do not swear in the name of idolatry. Even though they mention the idolatry, they intend for the Creator of Heaven and earth, but they join His name with something else. We do not find that Lifnei Iver applies to this; they are not commanded about Shituf (saying that Hash-m is a plurality). All permit buying and selling with Nochrim without partnersip, except for the day of their festival.

Magen Avraham (2): R. Aharon Sason says that sometimes it is forbidden to accept the oath.

Chasam Sofer: It is forbidden when it is not like saving from the Nochri. Regarding the trinity, a Ben Noach is killed for Avodas Shituf.

Eshel Avraham (2): The Nochri swears in the name of Hash-m and their prophets. This is forbidden for a Yisrael. If he joins (Hash-m) with idolatry, it is forbidden for a Nochri.

Machatzis ha'Shekel (2 DH Yischayev): Me'il Tzedakah (22) explains that a Nochri may not join something with Hash-m, just like Yisrael are commanded. Tosfos said only that they do not mention idolatry in the oath. Even if they refer to idolatry when they mention the Creator (for they believe that there are other parts to Hash-m), they only mention the Creator's name. A Yisrael may not equate a created being with the Creator; a Nochri is not commanded about this, so Lifnei Iver does not apply. Sha'ar Efrayim explains that Shituf contradicts the Nevu'ah of Moshe, and Nochrim are not commanded about this.

Kaf ha'Chayim (20): The Isur does not apply to Yishmaelim, for they swear only in the name of Hash-m.

Shulchan Aruch (CM 176:51): One may not become a partner with an idolater. If he became a partner, and the Nochri was obligated to swear to him, he may accept the oath.

Beis Yosef (182 DH Nimtza): Many Rishonim permit accepting the oath. Hagahos Ashri forbids becoming a partner. The Rosh and Avi ha'Ezri permit nowadays. The Rambam permits even when Nochrim swore in the name of idolatry; it is only Midas Chasidus to refrain.

Bedek ha'Bayis: Teshuvas ha'Rosh (18:11) says that if one sees that the Nochri is resolved to swear, there is no benefit to make him swear. Orchos Chayim brings from the Rashba (7:302) that Nochrim nowadays swear by the four (disciples of their Messi'ah), so a Yisrael may make them swear, and "Lo Yishma Al Picha" does not apply.

SMA (92): It is Midas Chasidus to exempt him from the oath.

Kaf ha'Chayim (156:21): Similarly, if one sees that his Yisrael opponent is about to swear falsely, he should exempt him from swearing.

Mishneh l'Melech (Hilchos Melachim 10:7 DH Shuv): In the Yerushalmi, R. Yonah and R. Acha say that a Nochri must fulfill his oath. Perhaps he need not fulfill vows, only oaths, for transgressing an oath is like blasphemy. Avimelech swore to Avraham and Yitzchak. Chalilah that they made him swear in the name of idolatry and transgress "v'Shem Elohim Acherim Lo Sazkiru." If a Nochri need not keep oaths, why did they accept it?