Is $\Bbb Z_p^2$ a Galois group over $\Bbb Q$?

There's indeed no continuous surjective homomorphism $\hat{\mathbf{Z}}^\times\to\mathbf{Z}_p^2$ for any prime $p$.

Indeed, we have $\hat{\mathbf{Z}}^\times\simeq\prod_p(\mathbf{Z}_p\times\mathbf{Z}/q_p\mathbf{Z})$. Any homomorphism into $\mathbf{Z}_p$ has to be trivial on $\mathbf{Z}_\ell$ for any $\ell\neq p$ and on $\mathbf{Z}/q_p\mathbf{Z}$ for all $p$. By continuity, it therefore factors through $\mathbf{Z}_p$. Since there's no surjective continuous homomorphism $\mathbf{Z}_p\to\mathbf{Z}_p^2$, we're done.

Tags:

Galois Theory

Profinite Groups

Field Theory

Extension Field

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