Like M. Emerton, Pierre's question makes me reflect on what most departments actually do to enforce common knowledge among (future) working mathematicians: their qualifying exams. I hope that most departments do have qualifying exams, with set syllabi, and that faculty take those exams seriously to make sure students really do learn enough of the set topics.

In my department, we try to be as explicit as we possibly can about what we require of our graduate students, so for instance you can click here for an official list of qual syllabi, and you don't have to scroll down very far to see that Lagrange's Theorem is right there on the list. The idea that a "respectable specialist in probability theory" disavowed any knowledge of Lagrange's theorem is disturbing to me, and I'm not willing to write it off so quickly. (Keep in mind though that people, being people, sometimes ask stupid questions during talks or forget things that on a better day they would know cold -- I have certainly done such things in departmental seminars and colloquia.) It makes me wonder whether this specialist really went to a graduate program which took this stuff seriously...and not, say, Cornell University, according to Thierry's distressing comment. (Seriously, for a very good department they have what looks to be a very bad approach to educating their graduate students.)

About what you can forget in other fields of mathematics, however basic...this is an interesting question. From my interactions with research mathematicians I have encountered quite a broad spectrum in terms of how much knowledge / awareness they have of fields other than their own. It's a fun thought experiment (read: don't do it!) to contemplate what would happen if you gave the graduate student quals to the faculty members instead. I myself am relatively fortunate to work in a "composite" field like arithmetic geometry which necessarily draws on multiple basic areas: there is of course no arithmetic geometry qualifying exam, but my knowledge in this subject makes the algebra and topology exams look pretty trivial to me. I would do better on the complex analysis exam if it covered fancier material -- sheaves and manifolds rather than, say, Rouche's Theorem -- but I could still pass it. The real analysis exam looks hard to me: I don't like my chances without studying for it. But my proclivities in mathematical knowledge run more to the "broad and shallow" -- as those who have seen me answer questions on this site will know -- with the pleasant side effect that I can often plop myself down in "someone else's" seminar without getting completely lost...or rather no more lost than I would be if a visiting number theorist started filling up the board with Iwasawa theory.

One thing to keep in mind is that the colloquium is the primary opportunity for many grown up mathematicians to get exposed to any ideas outside of their narrow field. That's why it's so important to have departmental colloquia (and so frustrating when the talks are bad enough to discourage people rather than drawing them in). If there wasn't a colloquium in your department before and there is now, then if you can keep the probabilists coming, they'll probably remember Lagrange's Theorem eventually.

Finally, there is so much art and skill in giving a good colloquium talk -- and this skill cuts transversally across other kinds of mathematical ability. I can see how the number theorist's first sentence put people off, but that could have been as much of a mistake of delivery as content. If s/he had simply said "As you may know..." people would probably not have rolled their eyes. Then, depending upon how important it was to the rest of the talk, the speaker could give some skillful, but brief "reminders" about the structure of the Galois group as an inverse limit of finite discrete groups. But only if that information turns out to be really important. If not, it would be better to start with something else and say at the appropriate time, in passing: "Now the group of all field automorphisms of the algebraic numbers -- which number theorists call the absolute Galois group of $\mathbb{Q}$ -- is a big, uncountably infinite group. However, it carries a natural topology with respect to which it is compact and totally disconnected, and this topology is important in the study of...."