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Katie there are two possibilities in your case.Firstly , what we very often see is called KERATOLYSIS EXFOLIATIVA.This is a benign condition where in we see peeling of skin esp in winters or because of dry skin
.There is no itching in this .And what you require is a good moisturizer daily 3-4 times and a good moisturizing soap to wash your hand with.
The second possibility is ALLERGIC CONTACT DERMATITIS
,where in we see uneven peeling of skin with itching .This is less likely in your case .One requires here topical steroids
The least possibility is a fungal infection
which usually involves only one hand ,and is associated with itching.
I would advise you to consult a dermatologist
who can clinically see and diagnose it best.