This is what I was missing. But the paper also says σ is an automorphism of Q(x_1, ... x_n). Which is weird, since I thought Q(x_1, ... x_n) was a subfield of the reals (not a field of rational functions). So I still don't get what's going on.
Sigh, I think I've forgotten more since school than I thought.
No field of rational functions here, that's waaaay off given what Stillwell intends to do.
Also, subfields of reals are a bit restrictive, don't you think?
Right-o!
> Now sigma is supposed to be an automorphism on that subfield. Which means sigma does take scalar values as arguments, contrary to what stablemap said...
There's that ol' abuse of notation going on here. The first sigma is just a permutation on the roots: {x_i} -> {x_i}. In particular, this skinny sigma is not defined on scalars.
It embiggens into a second sigma that's your automorphism: Q({x_i}) -> Q({x_i}). This fat sigma now maps scalars.
Now identify the first and second sigmas, and the abuse is complete.