Gemara

All agree that Matan Torah was on Shabbos. It says "Zachor Es Yom ha'Shabbos Lekadsho", like it says "...Zachor Es Yom ha'Zeh." Just like the latter refers to the day itself [that it was said], also the former;

Nazir 3b: R. Shimon holds that "from wine and strong drink Yazir" forbids to a Nazir wine of a Mitzvah, just like regular wine.

Suggestion: 'Wine of a Mitzvah' is wine of Kidush and Havdalah.

Rejection: (This cannot be.) One is bound (to drink it) by an oath (from Har Sinai)!

Pesachim 106a (Beraisa): "Zachor Es Yom ha'Shabbos Lekadsho" - one should be Zocher (mention) Shabbos (i.e. say Kidush) over wine.

117b (Rav Acha bar Yakov): One must mention Yetzi'as Mitzrayim in Kidush ha'Yom. It says "Lema'an Tizkor Es Yom [Tzeischa me'Eretz Mitzrayim]," just like "Zachor Es Yom ha'Shabbos Lekadsho."

Berachos 33b (Rava): Havdalah is like Kidush. Even though one says it in Shemoneh Esreh, he says it again over wine.

Rishonim

Rambam (Hilchos Shabbos 29:1): It is a Mitzvas Aseh mid'Oraisa to be Mekadesh Shabbos with words. "Zachor Es Yom ha'Shabbos Lekadsho" - mention it with praise and Kidush. One must mention Shabbos when it enters through Kidush, and when it ends through Havdalah.

Ran (10a DH Basar): If one does not have enough money for Ner Chanukah and wine for Kidush, why does Ner Shabbos or Ner Chanukah override Kidush mid'Oraisa? I answer that it does not override, for one may say Kidush on bread. If not for Shalom Bayis or publicizing the miracle, the ideal Mitzvah of Kidush would override, like we say "remember it over wine when it enters."

Rosh (Pesachim 10:5): R. Yonah says that the primary Kidush mid'Oraisa is like it says "remember it over wine when it enters." Since some do not know to say Kidush, we say Kidush in the Beis ha'Keneses to be Motzi them mid'Oraisa. I disagree. "Kidush is only where one will eat" implies that elsewhere, it is not Kidush at all! Rabah says so. Shmuel asked why we say Kidush in the Beis ha'Keneses. He did not hold that people are Yotzei mid'Oraisa through it!

Question: Since Nezirus forbids wine that is not a Mitzvah, it also forbids wine of a Mitzvah! Why did the Gemara say that Nezirus does not forbid wine of Kidush and Havdalah, due to the oath from Sinai?!

Answer (Tosfos Nazir 4a DH Mai): R. Tam says that the text should say (incredulously) 'is one bound by an oath (about these)?' Even though Kidush is mid'Oraisa, the obligation to say it over wine is not.

Rebuttal #1 (Ra'avan Sof Teshuvos, DH Hiksheh): Our text is correct. A Neder or Nezirus forbids Mitzvos only if one specified!

Rebuttal #2 (Or Zaru'a 2 Shabbos 25): If one swore to drink wine, Nezirus forbids wine because he can permit his oath. Nezirus does not take effect on wine of Kidush and Havdalah, for the oath of Sinai cannot be permitted.

Poskim

Shulchan Aruch (OC 271:1): When one comes home [from the Beis ha'Keneses on Friday night], he should eat immediately.

Magen Avraham (1): Tikunei Shabbos and Maharil (163) say to say Kidush before night. The Rama mi'Pi'ano (2) says that if one is not hungry to eat, he may wait, for he already mentioned Shabbos at its entrance in the Beis ha'Keneses. The Rambam and Tosfos (Sukah 38a DH Mai) say that mid'Oraisa, Kidush is without wine. Chachamim enacted Kidush on wine. I say that mid'Oraisa, one is Yotzei Kidush in Tefilah. It says "Zachor Es Yom ha'Shabbos", and he mentioned it! Chachamim enacted Kidush where one eats (OC 269:1). Also Tosfos connotes like this. He challenges R. Tam, who says that one without wine does not say Kidush at all. Would Chachamim uproot Kidush mid'Oraisa?! Rather, it suffices to be Yotzei in Tefilah. Also, we say "even though he was Mekadesh in Tefilah [and was Yotzei, he is Mekadesh also over a cup]." Some say that also Havdalah is mid'Oraisa (Magid Mishneh 29:1), and we say (Berachos 33a) that initially Chachamim enacted it [only] in Tefilah. The Rambam and Tosfos in Pesachim (106b DH Mekadesh, citing the Yerushalmi) connote like this.

Kaf ha'Chayim (3): Even an ignoramus who does not pray is Yotzei Kidush mid'Oraisa through hearing Birkas me'Ein Sheva (Magen Avos) from the Shali'ach Tzibur.

Eshel Avraham (1): Mid'Oraisa, one is Yotzei Kidush through Tefilah. Therefore, if there is a Safek whether or not he made Kidush, he need not make Kidush again, for we are lenient about a Safek mid'Rabanan. Also, a minor who reached the age of Chinuch can be Motzi an adult who prayed (Mishbetzos Zahav 1).

Mishnah Berurah (2): I have great doubts about this. We hold that Mitzvos require intent. People do not normally intend to be Yotzei the Mitzvas Aseh of Zachor through Tefilah, since they do not have wine or bread at the time, and they can say Kidush on them afterwards with a Berachah, like is proper. It is better to be Yotzei the Torah Mitzvah then, than to be Yotzei in Tefilah without a cup and not where he will eat. I gave other reasons to question this ruling in the Bi'ur Halachah. Therefore, one should refrain from being Yotzei Kidush through a minor. Even if he will say each word with him, one should not be Yotzei through him unless also the adult has bread or wine in front of him at the time of Kidush. Kidush of Yom Tov is only mid'Rabanan, so if one has only one cup [of wine], he leaves it for Shabbos, which is mid'Oraisa, and on Yom Tov he says Kidush over bread.

R. Akiva Eiger: The Ran holds that mid'Oraisa, Kidush is on wine. When Yom Kipur falls on Shabbos. Chachamim did not enact to say Kidush. This proves that one is Yotzei mid'Oraisa in Tefilah! However, the Rosh suggested that Kidush in the Beis ha'Keneses is to be Motzi people mid'Oraisa. This shows that they are not Yotzei in Tefilah! It is difficult to say that Kidush is for those who do not know how to pray. Also, one is Yotzei even through saying "good Shabbos" without praying!

Bi'ur Halachah (DH Miyad): The Ran and Rashba hold that Kidush on wine or bread is mid'Oraisa. Also the Rashba says so. The Rosh clearly holds that one is not Yotzei mid'Oraisa in Tefilah, and even not through Kidush not where one will eat. R. Yonah disagrees. How can the Ran, Rashba and Rosh explain "even though one was Mekadesh in Tefilah, he is Mekadesh over a cup"?! They must say that the Gemara learns from "remember it over wine" that one is not Yotzei through Tefilah at all.

Bi'ur Halachah: How is one Yotzei Kidush in Tefilah? One must mention Yetzi'as Mitzrayim, like it says in Pesachim! (One must mention Yetzi'as Mitzrayim every night. On Shabbos, one must mention it n Kidush.) The Rif and Rosh bring this. We do not mention Yetzi'as Mitzrayim in Tefilah at night. This is not difficult for the Rambam. He wrote that one is Yotzei through words, i.e. if he also mentions Yetzi'as Mitzrayim. He wrote like this in Sefer ha'Mitzvos 31, and connotes that this is mid'Oraisa. (However, it is astounding that he omitted this in the Mishneh Torah.) How can we answer for the Magen Avraham and the Acharonim who cited him?! The Minchas Chinuch rejected the Magen Avraham for this reason. I answer that the Magen Avraham holds that it is a mere Asmachta to mention Yetzi'as Mitzrayim. The Gezeirah Shavah is needed for the Drashah in Shabbos 86b. Alternatively, he holds that one is Yotzei through mentioning Yetzi'as Mitzrayim in Emes ve'Emunah. We say that Hashkivenu [the following Berachah] is like a continuation, so it is not a Hefsek. The first three Berachos of Shemoneh Esre are not a Hefsek, for they are praise of Hash-m. However, this is difficult. Perhaps one is Yotzei Kidush through saying "v'Shamru Bnei Yisrael Es ha'Shabbos..." right after Hashkivenu, which is not a Hefsek. This is another reason not to rely on Tefilah for Kidush mid'Oraisa.

Dagul me'Revavah: If a woman did not pray on Friday night, perhaps a man cannot be Motzi her in Kidush if he was Yotzei mid'Oraisa through Tefilah. Normally, one who was Yotzei can be Motzi. However, perhaps this is only due to Arvus (all Yisraelim are responsible for each other). The Rosh (Berachos 3:13) says that Arvus does not apply to women, therefore they are Motzi only one whose obligation is only mid'Rabanan. Perhaps she is not an Arev (guarantor) for others, but men are Arevim for her. This requires investigation.

Divrei Chachamim (226): Ha'Gaon R. C.P. Scheinberg Ztz"l and ha'Gaon R. Y. Kaminetsky Ztz"l say that if one intends not to be Yotzei through Tefilah, surely he can be Motzi his wife later.

Chasam Sofer (on Magen Avraham 271:1): In any case one should not intend to be Yotzei in TefilahError! Reference source not found., so he can be Yotzei mid'Oraisa ideally, over wine and where he will eat. One should intend to be Yotzei on Yom Kipur, even if it is not Shabbos.

Hilchos Shabbos b'Shabato (1 p.350): Ha'Gaon R. S.Z. Auerbach Ztz"l says to intend to be Yotzei in Tefilah. Ha'Gaon R. Y.S. Elyashiv Ztz"l says to intend 'I do like Chachamim intended.'

Teshuvos v'Hanhagos (1:274): One should intend 'I am Yotzei in Tefilah if this is the ideal Mitzvah.'