  In this essay, we see how prime cyclotomic fields (cyclotomic fields obtained
by adjoining a primitive p-th root of unity to Q, where p is an odd prime) can
lead to elegant proofs of number theoretical concepts. We namely develop the
notion of primary units in a cyclotomic field, demonstrate their equivalence to
real units in this case, and show how this leads to a proof of a special case
of Fermat's Last Theorem. We finally modernize Dirichlet's solution to Pell's
Equation.
