$ \mathbb{T} \stackrel{\text{def}}{=} \{ z \in \mathbb{C} : |z| = 1 \}$
$\widehat{\mathbb{T}} \stackrel{\text{def}}{=} \text{Hom}(\mathbb{T},\mathbb{T})$
To show that $\widehat{\mathbb{T}}$ can be identified with $\mathbb{Z}$, we need to construct a 1-1 correspondence $\widehat{\mathbb{T}}$→$\mathbb{Z}$. Define such a correspondence as follows:
$$ \phi :\mathbb{Z}→\widehat{\mathbb{T}}$$
by $$ \phi(n)= β_n $$where $ β_n$(α)=$α^n $ for all $ α \in \ \mathbb{T} $ .
I know that $\mathbb{T} $ ≅$\mathbb{R}/\mathbb{Z }$ by $γ:\mathbb{R}/\mathbb{Z } → \mathbb{T } $ that maps t to $e^{2πit}$.
I'd going to show :
every $ β \in\ \widehat{\mathbb{T}} $ gives rise to an element in $\mathbb{Z }$
can you help me?