How to compute degree of $f: S^1\to S^1$ defined as $e^{ix}\mapsto e^{iax}$?
It is known that $f$ induces a vector space isomorphism $H^1(f): \mathbb R\to \mathbb R$ which is multiplication by a number that we call degree $(f)$.
The following theorem is also known to me:
If $M,N$ are oriented, compact, connected, $n-$ dimensional manifolds, $f:M\to N$ is smooth map such that $q\in N$ is a regular value of $f$ (i.e., derivative of $f$ is surjective for all $p\in f^{-1}(q)$), then
$a) f^{-1}(q)$ is a finite set. That is, $f^{-1}(q)=\{p_1,...,p_k\}$ for some $p_i\in M$.
$b)$ degree($f)=\sum_i\epsilon(p_i)$, where $\epsilon(p_i)= 1$, if derivative $Df_{p_i}$ preserves orientation; and $\epsilon(p_i)=-1$ otherwise.
I tried the following:
$1)$ If $1$ is a regular value, then $f^{-1}(1)=\{p_1,..., p_k\}$. for $p_i\in S^1$.
$f^{-1}(1)=\{e^{ix}\in S^1: e^{iax}=1\}\implies \cos ax=1,\sin ax=0\implies ax=2n\pi \forall n\in \mathbb Z$.
Case $1:$ Suppose that $a$ is a non integer real number.
$f^{-1}(1)$ is an infinite set. So degree doesn't make sense.
Case $2:$ $a$ is an integer.
In this case, $f^{-1}(1)$ is finite.
$2)$ I don't understand how to calculate $\epsilon(p_i)$'s here: if it were $f(x,y)=(\cos ax, \sin ax)$, then no problem. But it is, $f(e^{ix})= e^{iax}$ so how to compute its differential to know if it is orientation preserving or not, i.e., to know $\epsilon(p_i)$?