I know this question has already been answered many times elsewhere. I now formally the definition of a tensor product, however I do not get the point of it: why is it used ans how to think of it?
To be a bit more precise, I bumped into a text considering $X=(\mathbb{Z}/N\mathbb{Z})^\times \otimes \mathbb{Z}/2\mathbb{Z}$, and stating that the characters of $X$ are exactly the quadratic characters of $(\mathbb{Z}/N\mathbb{Z})^\times$. Why is that so? (I guess I have no idea because I do not understand how to compute with such tensor products, define characters, etc.)