The author is defining a $ℚ$-monomorphism $i:ℚ(a)→ℂ$, where $a$ is the real cube root of $2$.
Every element of $ℚ(a)$ is of the form $p+qa+ra^2$, where $p, q, r ∈ℚ$, and $$i(p+qa+ra^2)=p+qwa+rw^2a^2$$ where $w=exp(2πi/3)$
Why is it that such automorphism will fix the rational numbers?
I would really appreciate any help/thoughts!
It follows directly from the definition of the map $i$ that $i(p)=p$ for all $p \in \mathbb Q$: just take $q=r=0$.
The more interesting question is why the map $i$ is well defined, or equivalently, why every element of $\mathbb Q(a)$ has a unique expression of the form $p+qa+ra^2$ with $p,q,r \in \mathbb Q$.