I started reading about the reciprocity law for polynomials over a finite field, and I came across the definition of the d-th power residue symbol:
Let $\Bbb F$ be a finite field with $q$ elements, $q=p^f$ ($p$ is the characteristic of $\Bbb F$), and let $A = \Bbb F[T]$. Let $P\in A$ be an irreducible polynomial, $d$ a divisor of $q-1$ and $a \in A$.
Definition: If $P$ does not divide $a$, let $(a/P)_d$ be the unique element of $\Bbb F^*$ such that $$a^{\frac{|P|-1}{d}}=\left(\frac{a}{P} \right)_d \pmod P$$
If $P|a$ define $(a/P)_d=0$. The symbol $(a/P)_d$ is called the d-th power residue symbol.
My question is: How do we know that $(a/P)_d$ is in $\Bbb F^*$ ? Why can't it be just any element of $(A/PA)^*$?