Can you plow a field with one tire chain?
 

Because the wheel outside of the furrow loses traction as opposed to the one in the furrow, should I add a chain just on the side of the wheel outside of the furrow? (I only have one tire chain, I don't have a pair) Would that effect the soil of even the tractor in someway if I were to do that?

Chain
 

Put the chain on the furrow side and add wheel weights to the land side and you won't have any problems.

Adding a chain to the land side will gain nothing, as all your weight is on in the furrow. ;-)