Help for $U = T^2$ has an $F$ distribution with 1 numerator and $v$ denominator degre

If $T$ has a $t$ distribution with $v$ degrees of freedom, then $U = T^2$ has an $F$ distribution with 1 numerator and $v$ denominator degrees of freedom.

First, I set $$T = \frac{Z}{\sqrt(W/v)}$$, where $T$ is t distribution with $v$ df, W is a chi-squared distributed variable with v df, and Z has a standard normal distribution.

Then, I set $$T^2 = \frac{Z^2}{(W/v)}$$...

My question is that can I claim $Z^2$ is a chi-squared distributed variable with 1 df... If so, how do we know? Also, how do we know or show Z and W are independent?

Thanks a lot