Learn Russian: Мы шли широким полем. - We were walking through the wide field.

This is the Bare Instrumental of Route. Russian uses the instrumental case without a preposition to indicate the physical space or path you are moving through. It acts just like the English 'through' or 'along' in this context.

Could I say по широкому полю or через широкое поле instead?

Yes, both are grammatically correct. По широкому полю (dative) focuses on the surface you are moving along, while через широкое поле (accusative) emphasizes crossing it from one side to the other. The bare instrumental широким полем is slightly more literary and focuses on the field as the surrounding environment of your journey.

Since a field is a wide, open space, why use the unidirectional verb шли instead of the multidirectional ходили?

Using шли implies you were making a single continuous trip in one general direction, likely toward a specific destination. If you had been wandering aimlessly around the field, you would use the multidirectional ходили.

Why does the adjective широким end in -им rather than the typical neuter instrumental ending -ым?