Q. Were the Magi Jewish? I’m thinking they must at least have been students of the Jewish Scriptures (in particular, Daniel and Micah). Otherwise, they wouldn’t have known or cared that a king of the Jews would have been born around that time. They certainly wouldn’t have cared about it enough to have made the trip to see him and had a desire to worship him. I can’t see how they could have derived all of this information from some kind of astrology.
You’re right that there was likely a Jewish influence on the Magi while they were still in “the east” (whether that means Babylon or Persia or somewhere in that area). Not all of the Jews who were taken to exile by the Babylonians returned under the Persians. For many centuries afterwards there was a flourishing Jewish community in Mesopotamia. For example, that’s where the so-called Babylonian Talmud originated in Late Antiquity (though the term “Babylon” was archaic by then). There were also distinct scribal and devotional practices that developed in this community that eventually had to be reconciled with the different ones that developed in Palestine. So the continuing Jewish community in Mesopotamia left us much evidence of its existence and activity.
Indeed, we may see the visit of the Magi as another such evidence. They were not Jews themselves; their name indicates that, at least initially and by profession, they were “magicians” and followers of Zoroastrianism. However, the term “magic” at the time included alchemy and astrology, the forerunners of chemistry and astronomy, so we should see these men as scholars of both religion and science. It’s not hard to imagine that they would have taken an interest in the religious thought and traditions of the Jews, who, like them, were monotheists. As you say, they would likely have read the Jewish Scriptures, and they could indeed have developed a Messianic expectation by reading the prophets.
We don’t know exactly what the “star” was that they saw “in the east” (i.e., while they were still in the east) or “when it rose.” (Either translation is acceptable.) A little later in Matthew’s account of their visit, after the Magi have stopped at Herod’s court and been directed to Bethlehem, there’s the puzzling statement that “the star they had seen when it rose went ahead of them until it stopped over the place where the child was.” So this wasn’t just a star that appeared in the sky in a new position, or was first recognized there, with some astrological significance. This is some kind of active agent of divine guidance. But we aren’t able to say much more about it than that.
What we can say, however, is that Matthew has a great interest in these Magi as Gentiles who have come to worship the Messiah. In the episode just before this one, an angel tells Joseph that Mary “will give birth to a son, and you are to give him the name Jesus, because he will save his people from their sins.” But ironically, the very next thing that happens is that Babylonians or Persians come and worship Jesus, while “his people”—and specifically Herod, who considers himself the “king of the Jews”—try to kill Jesus, and he has to flee to Egypt. (Where, by the way, there was another longstanding Jewish community. So there are two allusions to the Diaspora, and thus to Jewish influence on the surrounding nations, in this one episode.) This is in keeping with one of Matthew’s large themes, the inclusion of the Gentiles in the community of the Messiah’s followers. For example: “Many will come from the east and the west and take their places at the feast with Abraham, Isaac and Jacob in the kingdom of heaven.”
So the Magi were not Jews, but under Jewish influences they did come to expect the Messiah. And then they came to worship the Messiah.