Let p be a
prime, and denote the
field of
p-adic numbers, as usual, by . Then the
Galois group, where denotes the
algebraic closure of , is prosolvable. This follows from the fact that, for any finite
Galois extension of , the Galois group can be written as
semidirect product, with cyclic of order for some , cyclic of order dividing , and of -power order. Therefore, is solvable.[1]