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If I were to buy a little bit of an adjacent field from a farmer to
extend my garden would I therefore need planning permission to grow
anything but grass ?.

If you were changing it from agricultural use you are likely to need
pp for change of use, which may be refused. Local Plans in rural areas
usually set a boundary between permitted housing development and
designated greenbelt. There are well documented cases of people buying
designated agricultural land, extending their designated "housing plot"
into it, falling foul of their Local Plan and having to return their
garden to pasture.

The idear might be, it seems building land is expensive, agricultural
land is cheap, buy a home with a small garden next to agricultural
land, extend size of garden.

Very naive. Even if you could persuade a farmer to sell, it wouldn't
be at agricultural land rate. Farmers aren't stupid, they know what
extra land adds to the value of a domestic property, and would set the
price accordingly.

Janet