Please help clear doubt about algebraic extension

Question

This is from Artin's Algebra:

I am very confused here: I can't figure out if they are contrasting para 1 above with prop 15.2.8 or generalizing para 1 in prop 15.2.8, or what?

Also, I can't see how $\Bbb Q(\alpha)$ and $\Bbb Q(\beta)$ are the same.

Answer 1

From my point of view, the observation in § 1 is contrasting with Prop. 15.2.8, since $\mathbf Q(\alpha)$ and $\mathbf Q(\alpha+1)$ are more than isomorphic: they're equal, yet $\alpha$ and $\alpha+1$ do not have the same irreducible polynomial. The difference lies in the fact that the map $x\rightsquigarrow x+1$ on $\mathbf Q(\alpha)$ is a field isomorphism, but not a $\mathbf Q$-isomorphism.

As to your second question, note that $\alpha+1\in\mathbf Q(\alpha)$, whence a first inclusion, and that $ \alpha=(\alpha+1)-1$, whence the reverse inclusion.