Why do we use "a" in the first sentence?
Kevin works in a hospital. He operates on people.
Sarah looks after patients in hospital.
Note: the two examples from the same book: English Grammar in Use
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In British English, in hospital means "under hospital care" or "receiving hospital care". For that reason, we would say:
She looks after patients in hospital.
That is, She looks after patients (who are) under hospital care.
But when we want to refer to the place as distinct from the care provided there:
Kevin works in a hospital.
Kevin is not under hospital care. He works there.
In American English, the phrase that means "under hospital care" is in the hospital.
She had a car accident.
-- Is she going to be OK?
She's in the hospital, with broken ribs and lacerations.
When speakers of AmE say in the hospital in such contexts, they do not mean a particular hospital or an aforementioned hospital. They mean under hospital care.
Kevin works in a hospital. Kevin is an employee of that hospital.
Sarah looks after patients in hospital. Not at home, or other place.