Well, now I feel silly -- on looking through Wieslaw again, I see he does give examples of non-discrete, non-straight fields, just not in that section. For instance, take two absolute values on the rationals; the topology they generate together still make the rationals a topological field, is not discrete, and is obviously finer than either of the ones you started with. Since it isn't even minimal, it can't be straight.

I'm not going to accept my own answer on this since this example presumably isn't complete, and I'd like a complete example. Which might well be in here too, if I keep looking... (Is the completion of this again a field? If so that should work, but I'd need to check if that's true.)

Edit: Nope, the completion of this isn't a field, so I'm still lacking for a complete example.

Well, now I feel silly -- on looking through Wieslaw again, I see he does give examples of non-discrete, non-straight fields, just not in that section. For instance, take two absolute values on the rationals; the topology they generate together still make the rationals a topological field, is not discrete, and is obviously finer than either of the ones you started with. Since it isn't even minimal, it can't be straight.

I'm not going to accept my own answer on this since this example presumably isn't complete, and I'd like a complete example. Which might well be in here too, if I keep looking... (Is the completion of this again a field? If so that should work, but I'd need to check if that's true.)

Edit: Nope, the completion of this isn't a field, so I'm still lacking for a complete example.

Edit again: OK, I'm now pretty sure Wieslaw gives one, so the answer is no. Wieslaw gives an example of a complete normed field which is not given by any absolute value (here "normed" means instead of |ab|=|a||b|, we only require |ab|≤|a||b| and |-a|=|a|). Furthermore, he shows given a power-multiplicative norm, you can find an absolute value that generates a coarser topology (so if a power-multiplicative norm wasn't equivalent to an absolute value, it isn't minimal). (Here power-multiplicative only means we require |aⁿ|=|aⁿ| for positive n, not for negative n.) And after a bit of staring at his example, I'm pretty sure it's power-multiplicative. So unless anyone can show that I've missed something, I'm going to consider this one closed.

Well, now I feel silly -- on looking through Wieslaw again, I see he does give examples of non-discrete, non-straight fields, just not in that section. For instance, take two absolute values on the rationals; the topology they generate together still make the rationals a topological field, is not discrete, and is obviously finer than either of the ones you started with. Since it isn't even minimal, it can't be straight.

I'm not going to accept my own answer on this since this example presumably isn't complete, and I'd like a complete example. Which might well be in here too, if I keep looking... (Is the completion of this again a field? If so that should work, but I'd need to check if that's true.)

Well, now I feel silly -- on looking through Wieslaw again, I see he does give examples of non-discrete, non-straight fields, just not in that section. For instance, take two absolute values on the rationals; the topology they generate together still make the rationals a topological field, is not discrete, and is obviously finer than either of the ones you started with. Since it isn't even minimal, it can't be straight.

I'm not going to accept my own answer on this since this example presumably isn't complete, and I'd like a complete example. Which might well be in here too, if I keep looking... (Is the completion of this again a field? If so that should work, but I'd need to check if that's true.)

Edit: Nope, the completion of this isn't a field, so I'm still lacking for a complete example.