Suppose you have a prime $p$ (not necessarily odd) and you have $\tau$ defined by $p^\tau \| k$ for some integer $k$. Then you define \begin{equation} y = \begin{cases} \tau + 1 &, p > 2 \textrm{ or } p = 2 \textrm{ & }\tau = 0\\ \tau + 2 &, p = 2 \textrm{ & } \tau > 0 \end{cases} \end{equation}
Then we can use chapter 6, section 5 of Vinogradov's Elements of Number Theory to say that there are, at least for odd primes, $\phi(p^{\tau + 1})/(k, \phi(p^{\tau + 1}))$ $k$th power residues, modulo $p^y$.
1) What does the condition $p^\tau \| k$ give us? I could understand if we were just stripping multiples of $p$ off $k$ so that we could essentially work with $1 \leq k \leq p$ but this doesn't seem to be for that.
2) I'm guessing that the reason for question (1) tells us why we want $y$ to have that form, and I've got a rough feeling as to why we have that form for odd primes, but I'm really not seeing it for $p = 2$
3) The book goes on to say that the number of solutions to $x^k \equiv a \pmod{p^y}$ is $p^{y - \tau - 1}(k, \phi(p^{\tau + 1}))$ which, again, I see in the case for odd primes but for $p = 2$ I'm missing something (and most texts I've got access to seem to ignore power residues modulo $2^\alpha$.
4) It also then goes on to state without proof that if $a$ is a $k$th power residue modulo $p^y$ then it is a $k$th power residue modulo $p^t$ for all $t$, which I guess is also connected to some fundamental property about the definition of $y$ which I'm just being really stupid about.
Any help with these four questions would be appreciated :)
