Itchy, scaly skin is a sign of psoriasis and it can affect the armpits. My sister has psoriasis and she does experience armpit peeling sometimes. She has several prescription creams that she uses when it happens.
Psoriasis is caused by the immune system so there isn't a cure. Treating the symptoms is the most that can be done. We have noticed though that my sister's psoriasis worsens when she is stressed. So avoiding stress is a good idea.