Rosh Hashanah, Chapter 4, Mishnah 5. This mishnah discusses the order of the berachos of the Mussaf prayer on Rosh Hashanah.
The View of Rabbi Yochanan ben Nuri - The Order of the Berachos:
The mishnah opens: "Seder berachos: omer Avos u'Gevuros u'Kedushas HaShem" - the order of the berachos: one says Avos, Gevuros and Kedushas HaShem. These are them:
"Avos" - the berachah that we conclude with "Magen Avraham".
"u'Gevuros" - the berachah that we conclude with "Mechayeh HaMeisim".
"u'Kedushas HaShem" - this is "HaMelech HaKadosh", "v'kolel Malchuyos imahen v'eino tokea" - and he includes Malchuyos together with them and does not blow. This is the opinion of the first Tanna, that Malchuyos is included together with "HaMelech HaKadosh", all joined as one, and the shofar is not sounded at this point.
"Kedushas hayom v'sokea" - the berachah of "Mekadesh Yisrael v'Yom HaZikaron", and then the shofar is sounded.
"Zichronos v'sokea" - the berachah of "Zocher HaBris", and he blows.
"Shofros v'sokea" - the berachah of "Shomea kol teruas amo Yisrael b'rachamim", and he blows.
"V'omer" - and afterward one says the final three berachos that are recited in every Shemoneh Esrei:
"Avodah" - this is "Retzei", which concludes with "HaMachazir Shechinaso l'Tzion".
"V'hoda'ah" - this is "Modim", which concludes with "HaTov shimcha ul'cha na'eh l'hodos".
"U'virkas kohanim" - this is "Sim Shalom", which concludes with "HaMevarech es amo Yisrael bashalom".
"Divrei Rabbi Yochanan ben Nuri" - these are the words of Rabbi Yochanan ben Nuri. This is his view, and its significant point is that he includes Malchuyos in the berachah of Kedushas HaShem, "HaMelech HaKadosh", and does not sound the shofar at that time, but rather blows at the berachah of Kedushas HaYom.
Rabbi Akiva's Objection:
Rabbi Akiva said to him: "Im eino tokea l'Malchuyos, lamah hu mazkir?" - if he does not blow for Malchuyos, why does he mention it at all? For Malchuyos, Zichronos and Shofros, by their very definition, come together with the sounding of the shofar.
The View of Rabbi Akiva:
"Ela omer Avos u'Gevuros u'Kedushas HaShem" - rather, he says Avos, which concludes with "Magen Avraham"; Gevuros, which concludes with "Mechayeh HaMeisim"; and Kedushas HaShem, "HaMelech HaKadosh", which stands on its own.
"V'kolel Malchuyos im kedushas hayom v'sokea" - Malchuyos is attached to the berachah of Kedushas HaYom, and he says "Melech al kol ha'aretz, Mekadesh Yisrael v'Yom HaZikaron", which is the conclusion actually recited in the Shemoneh Esrei of Rosh Hashanah, and he blows - on account of the Malchuyos that are included in Kedushas HaYom.
"Zichronos v'sokea, Shofros v'sokea" - in order.
"V'omer Avodah v'hoda'ah u'virkas kohanim" - the final three berachos, as Rabbi Yochanan ben Nuri said.
In summary: The decisive difference between the two views is where Malchuyos is placed and where the shofar is sounded. According to Rabbi Yochanan ben Nuri, Malchuyos is included in Kedushas HaShem, "HaMelech HaKadosh", and no blowing takes place there, only at Kedushas HaYom. According to Rabbi Akiva, whose view we follow as the halachah, "HaMelech HaKadosh" is said on its own without blowing, while Malchuyos is attached to Kedushas HaYom and the shofar is sounded there, followed by Zichronos, Shofros and the remaining berachos.