Does $X_1,...,X_n$ being a random sample from $N(\mu,\sigma^2)$ $\implies \frac{\overline{X}-\mu}{\frac{s}{\sqrt{n}}}$ ~$t_{n-1}$?

Question

Does $X_1,...,X_n$ being a random sample from $N(\mu,\sigma^2)$ $\implies \frac{\overline{X}-\mu}{\frac{s}{\sqrt{n}}}$ ~$t_{n-1}$?

If so does the above imply that a standard normal divided by the square root of a $\chi^2$ variable with $a$ degrees, has a $t$ distribution with $a$ degrees of freedom?