What is the bit complexity of Gaussian eliminaton over $\Bbb F_q$? Given matrix $M\in\Bbb F_q^{n\times n}$ with rank $r$ what is the complexity of converting to row-echelon form?


  Is it $O(n^3\log q)$ or $O(n^3q)$ bit operations?


Technically $O(n^3)$ row operations in $O(\log q)$ bit words should be $O(n^3(\log q)(\log\log q)^2)$ bit operations in worst case when $q$ is prime. 

What I am confused about is


when $q$ is prime power 
incorporating complexity of modular operations.
The dependence on the field is via the cost of arithmetic operations over the field. Since numbers in $\mathbb{F}_q$ have bit size $O(\log q)$, the cost of each individual operation is polylogarithmic in $q$.