A little confusion on galois groups

Question

We know $Gal(Q(\zeta_n)/Q) \cong (Z/nZ)^*$ for $n\geq 2$.
Then $Gal(Q(\zeta_2)/Q) \cong (Z/2Z)^*$ this means $Gal(Q(\zeta_2)/Q)$ has one element, the identity automorphism. The map $\delta:Gal(Q(\zeta_2)/Q)\rightarrow Gal(Q(\zeta_2)/Q)$
where $\delta(a_0 + a_1\zeta^1)=a_0 - a_1\zeta^1$ is an automorphism that fixes Q by applying the property of ring homomorphisms.
QUESTION: Whys is it that $\delta$ is not an element of $Gal(Q(\zeta_2)/Q)$?

Answer 1

Hint: What is $\Bbb Q(\zeta_2)$? What is your $\zeta$? Write it out explicitly, and you will probably see your error.