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Prime fields are a very special case of finite fields, so unsurprisingly there are major implementation differences.

Those differences however should, as much as possible, be smoothed out in the user interface, and you are perfectly right in pointing out this inconsistency.

Fixing this is now tracked at

Similar tickets exist to make available methods more consistent for various implementation of finite fields:

Similarly there have been tickets to make some of the number field methods available for QQ too: