  We first show joint uniform distribution of values of Kloosterman sums or
Birch sums among all extensions of a finite field $\mathbb{F}_q$, for almost
all couples of arguments in $\mathbb{F}_q^\times$, as well as lower bounds on
differences. Using similar ideas, we then study the biases in the distribution
of generalized angles of Gaussian primes over function fields and primes in
short intervals over function fields, following recent works of Rudnick--Waxman
and Keating--Rudnick, building on cohomological interpretations and
determinations of monodromy groups by Katz. Our results are based on generic
linear independence of Frobenius eigenvalues of $\ell$-adic representations,
that we obtain from integral monodromy information via the strategy of
Kowalski, which combines his large sieve for Frobenius with a method of
Girstmair. An extension of the large sieve is given to handle wild ramification
of sheaves on varieties.
