Non-standard finite fields over I Delta(0)+Omega(1)

We consider residue fields of primes in the well-known fragment of arithmet ic I Delta(0) + Omega(1). We prove that each such residue field has exactly one extension of each degree. The standard proofs use counting and the Fro benius map. Since little is known about these topics in fragments, we looke d for, and found, another proof using permutation groups and the elements o f Galois cohomology. This proof fits nicely into I Delta(0) + Omega(1) usin g, instead of exponentiation, exponentiation module a prime.