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.