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Nov 4, 2015 at 8:25 vote accept Mark Grant
Nov 3, 2015 at 21:08 comment added Qiaochu Yuan @Mark: you can reduce to the line bundle case for this without the splitting principle. It follows from the fact that complexification commutes with taking exterior powers (which in turn follows from the fact that it's symmetric monoidal).
Nov 3, 2015 at 21:03 comment added Mark Grant This is great, thanks. I guess that, more generally, the first Chern class of the complexification of a real bundle of any rank is the Bockstein of its first Stiefel-Whitney class? Probably this follows from the splitting principle?
Nov 3, 2015 at 19:55 history answered Qiaochu Yuan CC BY-SA 3.0