Field extensions in CS A field is a set with two binary operations called addition and multiplication satisfying various axioms. Wikipedia article: Field_(mathematics)

A field extension is when you add a new element and then have to add all arithmetic combinations of that new element with the existing elements, e.g. adding i to the real numbers to get the complex numbers. If F is a subfield of K then K is called an extension field of F.

Technically you aren't adding new elements, but defining a new field which has a subset isomorphic to the orginal field, e.g. the complex numbers x+0i are identified with the real numbers x.

Wikipedia: Field_extension

Finite field extensions are useful in the construction of BCH codes which are codes over the finite fields of order q=pn which are extensions of the fields of order p where p is prime. These finite fields known as Galois fields GF(p) are extended to GF(q) by adding roots of polynomials that are irreducible (non-factorable) over GF(p).

Wikipedia: BCH_code

Where else in CS does the concept of extension fields arise ? A fundamental appearance in some subject would be more interesting than an occurence in some proof, but if it is used frequently in a common proof stategy then such a proof strategy would also be an interesting answer here.
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.