Chebyshevʼs bias in Galois extensions of global function fields

We study Chebyshevʼs bias in a finite, possibly nonabelian, Galois extension of global function fields. We show that, when the extension is geometric and satisfies a certain property, called, Linear Independence (LI), the less square elements a conjugacy class of the Galois group has, the more primes there are whose Frobenius conjugacy classes are equal to the conjugacy class. Our results are in line with the previous work of Rubinstein and Sarnak in the number field case and that of the first-named author in the case of polynomial rings over finite fields. We also prove, under LI, the necessary and sufficient conditions for a certain limiting distribution to be symmetric, following the method of Rubinstein and Sarnak. Examples are provided where LI is proved to hold true and is violated. Also, we study the case when the Galois extension is a scalar field extension and describe the complete result of the prime number race in that case.