How central to the theology of Judiasm is the Exile? Is it like Christianity where the divinity of Christ is necessary (IIRC), or Islam where one must practice all five pillars to be a muslim? Or is it so much fluff – you are free to take it or leave it?
Which “exile” are you talking about? Babylon, Egypt or the diaspora?
Are you talking about the Babylonian Exile following the destruction of Judah in 586 BCE? I am not aware of any particular necessary belief connected with it. (It isn’t hard to believe it, anyway, inasmuch as it is a historical event.)
If you are asking what impact it had on Judaism, the answer is lots. Different people actually date the creation of the the religion of Judaism to the return of the exiles from Babylon. (This would obviously not be an absolute claim, and I would not try to defend it, but it is a view that some hold.) Certainly, many of the expressions of Judaism in the post-exilic period were shaped by the experience in Babylon. In addition, the ability of the Jewish people simply to resist assimilation at that point established the precedent by which they were able to maintain their identity through various later exiles and diaspora. Beyond that, many of the exiles established a scholarly community that continued without interruption for 1500 hundreds of years, even when the people in Jerusalem were subjected to attacks, exiles, and even internecine warfare, thus providing a haven for Jewish thought and expression outside the conflicts of the Holy Land.
The diaspora, I guess. The current exile – aren’t they* still waiting for a messiah or something like that?
*Okay, that phrasing sounds pretty ugly. I apologize for that.
The concept of the Messiah goes back before the Babylonian exile; my reading indicates that it was originally just a descriptor for the king - “the annointed.” Later, as kings began letting them down, Jews began to look forward to (and prophesy of) a perfect king who would be God’s representative on Earth. Under Persian influence (and later Alexandrian influence), the concept of the Messiah grew to a savior, who would rescue the Jews, restore Jerusalem, and bring all the Gentiles to God’s kingdom. Some prophets (these are the verses seized on by Christians) spokea supernatural being who would bring God’s kingdom to Earth, resurrect the dead, make the lion to lie down with the lamb, etc.
The Messiah is indeed one of the principles of Jewish faith: “I believe with perfect faith in the coming of the moshiach, and though he may tarry, still I await him every day.” (Principle 12 of Rambam’s 13 Principles of Faith). It’s a mistake, however, to assume that this belief is common to all modern Jews, and a much greater mistake to attribute it to the diaspora.
Judaism as a whole isn’t really dogmatic enough to compare with Christianity or Islam; law, not belief, is the central feature of Jewish life, and Jewish culture persists where neither law nor dogma holds sway.