Non-standard finite fields overIΔ0+Ω1

We consider residue fields of primes in the well-known fragment of arithmeticIΔ0+Ω1. We prove that each such residue field has exactly one extension of each degree. The standard proofs use counting and the Frobenius map. Since little is known about these topics in fragments, we looked for, and found, another proof using permutation groups and the elements of Galois cohomology. This proof fits nicely intoIΔ0 + Ω1 using, instead of exponentiation, exponentiation modulo a prime.