Here are examples of scoring for points for fields when they're touching cities.

Each farm is treated completely independently. For the top left farm yellow scores 6 points. For the larger farm, yellow and red share the majority and each one scores 6 points. So yellow gets 12 points in total and red 6.

In this case there's only one big farm with 2 complete cities. The yellow has the majority, so it will score 6 points and the red 0 points.