where formula_8 is the field generated by a primitive formula_9-th root of unity. This tower of fields has a union formula_10. Then the Galois group of formula_11 over formula_12 is isomorphic with formula_1; because the Galois group of formula_14 over formula_15 is formula_16. In order to get an interesting Galois module here, Iwasawa took the ideal class group of formula_8, and let formula_18 be its formula_4-torsion part. There are norm mappings formula_20 when formula_21, and so an inverse system. Letting formula_22 be the inverse limit, we can say that formula_23 acts on formula_24, and it is good to have a description of this action.