I am working on the following exercise in a syllabus by Lenstra.
After having solved the first parts of the exercise, I am stuck at the last part of proving that $\mathbf{Q}^\times \times \widehat{\mathbf{Z}}^\times\to \{\pm 1\},(a,n)\mapsto \left(\frac{a}{n}\right)$ is continuous (and bimultiplicative).
Proving the continuity is equivalent to finding the preimage of $1$ and proving this is open. However, it seems to me that is very difficult to explicitly state what this preimage is.
Could someone please provide any help?