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Only capitalise the names of degrees if you're using the full title:
She is studying a BA in Ancient History and Archaeology.
Students with a history degree studied in the Department of History.
Types of degree are always lower case:
bachelor's
master's
postdoctorate
Here's how to write common degrees as abbreviations:
BA
BEng
BSc
MA
MEng
MPhil
MRes
MSc
PGDip
PGCert
PhD