I once spent a year in Cambridge, England. The local farmers used to harvest their crops—wheat, barley, corn, or whatever—and then when the stubble had dried out, they would burn it off. That quickly prepared the land for planting winter wheat, by removing the stubble and getting the remains into the ground, and I suppose as you say reduced the carbon.
It was an interesting sight, although things got pretty smokey right into town in the burning season. I don’t know if they still do it. I doubt it, because the greenhouse gas folks would probably be all over them if they did.
Gone with the fox hunts, I expect.
Said farming method dates back to at least Old Testament times. See Malachi 4:1.