Gowers mentions that, to him, an obvious argument would be: 23 x 22 != 21 x 25, since 2 divides 23x22, but 2 does not divide 21x25.
I don't follow the distinction - isn't the same true of my example, just swapping "2" for "11" or "23"?
I mean, given Gowers' standing I'm prepared to assume his premise is true, and I didn't follow the last argument (by comparison to other kinds of rings) at all so I'm guessing that it's the "real" reason why the theorem isn't obvious. But the preceding arguments made no sense to me at all.
I think we're all at a loss for a good definition of "obvious".