Relationship between Fisher Test and Student Test When performing linear regression with one variable, one can compute Fisher's test with value $F$, and derive Student's test $T=\sqrt{F}$.
When there is more than one variable, the relationship $T=\sqrt{F}$ no longer holds. Is there a relationship between $F$ and the different Student tests $T_i$ (one for each variable)? Perhaps a system of equations?
EDIT :
I am referring to the F test in Excel's linear regression report:

Note : crossposted here with no answer.

I'm not sure that this is what you mean, but the $T^2$ Hotelling test is a generalization of the $t$-test for multivariate normal means.
Suppose that $X \sim N(\mu, \Sigma)$, where the number of dimensions is $p$. The number of observations is $n$. The test statistic is $$T = n (\bar{X} - \mu)' \hat{\Sigma}^{-1} (\bar{X} - \mu)$$, which using SVD decomposition for $\hat{\Sigma}$ is, suppose that $S$ is a diagonal matrix of the eigen-values and $U$ is the eigenvectors of $\hat{\Sigma}$. You obtain
$$ T= n (U(\bar{X}  -\mu))' S^{-1} U(\bar{X} - \mu),$$ which can be viewed as sum of squared $t$-statistics.
The statistic is related to $F$ distribution by $\frac{n-p}{p(n-1)}T \sim F_{p, n-p}$.
