Suposse a random sample of size $n$ from a Nomal distribution $X_{i}\sim N(\mu,\sigma^{2})$, for the following random variables:
(1) $\frac{\overline{X}-\mu}{S/\sqrt{n}}\sim t(n-1)$ and (2) $\frac{(n-1)S^{2}}{\sigma^{2}}\sim\chi^{2}(n-1)$.
Their distribution does not depend on unknown parameters, so they are Pivotal Quantity,
But another way to obtain a Pivotal Quantity is by using the fact that the normal distribution is a distribution with location-scale parameters, and using $\hat{\mu}$ and $\hat{\sigma}$, from which we conclude that:
$\frac{\hat{\mu}-\mu}{\hat{\sigma}}$ and $\frac{\hat{\sigma}}{\sigma}$ are pivotal quantities for $\mu$ and $\sigma$ respectively.
¿It is possible to use this last reasoning to conclude that (1) and (2) are Pivotal Quantities?
For example:
I know that $\hat{\sigma^{2}}=\frac{n-1}{n}S^{2}$, I'm not sure about $\hat{\sigma}$, but some books consider it as $\hat{\sigma}=\sqrt{\hat{\sigma^{2}}}=\sqrt{\frac{n-1}{n}}S$, since $\frac{\hat{\mu}-\mu}{\hat{\sigma}}$ is Pivotal Quantity for $\mu$, we have:
$\frac{\hat{\mu}-\mu}{\hat{\sigma}}=\frac{1}{\sqrt{n-1}}\frac{\overline{X}-\mu}{S/\sqrt{n}}$ is Pivotal Quantity, but it doesn't seem to be the same as (1) except for some detail that I may not be able to see.