Injection from $(K\otimes_{\mathbb{Z}}\hat{\mathbb{Z}})^*\to \prod_p(K\otimes_{\mathbb{Z}}\mathbb{Z}_p)^*$? Suppose $K$ is a number field. The projections $\hat{\Bbb{Z}} = \prod_p\Bbb{Z}_p\to\Bbb{Z}_p$ give a map
$$(K\otimes_{\Bbb{Z}}\hat{\Bbb{Z}})^*\to \prod_p(K\otimes_{\Bbb{Z}}\Bbb{Z}_p)^*.$$
I want to show that this map is injective. It is clear to me that if $x\otimes (y_p)\in K\otimes_{\Bbb{Z}}\hat{\Bbb{Z}}$ is a unit, then it maps to $\prod_p 1\otimes 1$ if and only if $x=1$ and each $y_p=1$. But surely there are units that are not single tensors, and that is where I don't know how to proceed; I have very little intuition about either tensor products or $\hat{\Bbb{Z}}$.
I would prefer some explanation of the concepts, or of the problem I'm having, to a solution to my specific question.
 A: In fact, more is true: the ring homomorphism $K\otimes_{\Bbb Z}\hat{\Bbb Z} \rightarrow \prod_p(K \otimes_\Bbb Z \Bbb Z_p)$ is injective.
To see this, we first look at the case $K = \Bbb Q$. It is easy to see that every element of $\hat{\Bbb Q} = \Bbb Q\otimes_{\Bbb Z} \hat{\Bbb Z}$ can be written as a pure tensor (because a "common denominator" can be found for any finite sum of pure tensors), thus you should be able to understand this special case.
For general $K$, we fix an isomorphism $K \simeq \Bbb Q^d$ as $\Bbb Q$-vector spaces. We then have $$K\otimes_{\Bbb Z}\hat{\Bbb Z} \simeq K\otimes_{\Bbb Q}\Bbb Q \otimes_{\Bbb Z} \hat{\Bbb Z} \simeq \hat{\Bbb Q}^d$$ and also $$K\otimes_{\Bbb Z} \Bbb Z_p \simeq K\otimes_{\Bbb Q}\Bbb Q \otimes_{\Bbb Z}\Bbb Z_p \simeq \Bbb Q_p^d.$$
Together with the projection maps $K\otimes_{\Bbb Z} \hat{\Bbb Z}\rightarrow K\otimes_{\Bbb Z}\Bbb Z_p$ and $\hat{\Bbb Q}^d \rightarrow \Bbb Q_p^d$, we get a commutative square. The injectivity then follows from the above special case.

There is actually a more explicit description of $K\otimes_Z \hat{\Bbb Z}$. It is equal to the following set: $$\{(x_p)\in\prod_p (K\otimes_{\Bbb Z}\Bbb Z_p): x_p\in \mathcal O_K\otimes_{\Bbb Z}\Bbb Z_p, \text{almost all } p\},$$ where "almost all" means "all but finitely many". This is sometimes called a "restricted tensor product".
The group of units $(K \otimes_{\Bbb Z}\hat{\Bbb Z})^\times$ can be described similarly, by adding $^\times$ everywhere.
You may read the wiki page on adele rings or many books on the subject for more information.
