  Wedderburn showed in 1905 that finite fields are commutative. As for infinite
fields, we know that superstable (Cherlin, Shelah) and supersimple (Pillay,
Scanlon, Wagner) ones are commutative. In their proof, Cherlin and Shelah use
the fact that a superstable field is algebraically closed. Wagner showed that a
small field is algebraically closed, and asked whether a small field should be
commutative. We shall answer this question positively in non-zero
characteristic.
