Since $i$ is a map of rings, if $\tau$ were a map of rings, then $\tau\circ i$ would also be a map of rings. But the latter maps $1\in H^0(G_2)$ to $[G_2:G_1]$ which, in general, is not $1$.
Mariano Suárez-Álvarez
- 47.3k
- 14
- 147
- 264