Field extensions in CS
I used field extensions (and Tarksi's transfer principle) to prove lower bounds for linear degeneracy problems.  Using a field extension of the reals with formal infinitesimals simplified the construction of adversarial inputs.  (But see also Ailon and Chazelle's followup paper, which generalized my result without using field extensions.)

More generally, in the algebraic decision tree model (with real coefficients), the complexity of any problem with real inputs is identical to the complexity of the same problem where the inputs are drawn from any real closed field that contains the reals.