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Apr 1, 2014 at 16:15 comment added stankewicz You're right. I still think there's a way to say that if $Y$ was also Galois then there's a way to break it up into a composite cover with $X$ in the middle but I don't really have any evidence for that.
Apr 1, 2014 at 16:14 history edited stankewicz CC BY-SA 3.0
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Apr 1, 2014 at 12:35 comment added Ariyan Javanpeykar Nice example. I don't think this is an answer to the second question, though. In fact, the OP seems to ask whether $Y$ is a Galois cover of the projective line, as opposed to the composite cover $Y\to X\to \mathbf P^1$ being Galois. These are two different things, unless I'm misunderstanding...
Apr 1, 2014 at 11:18 history edited stankewicz CC BY-SA 3.0
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Apr 1, 2014 at 11:11 history answered stankewicz CC BY-SA 3.0