This paper describes a generalisation of the methods of Iwasawa Theory to the field F_∞ obtained by adjoining the field of definition of all the p-power torsion points on an elliptic curve, E, to a number field, F. Everything considered is essentially well-known in the case E has complex multiplication, thus it is assumed throughout that E has no complex multiplication. Let G_∞ denote the Galois group of F_∞ over F. Then the main focus of this paper is on the study of the G_∞-cohomology of the p~∞- Selmer group of E over F_∞, and the calculation of its Euler characteristic, where Possible. The paper also describes proposed natural analogues to this situation of the Classical Iwasawa λ-invariant and the condition of having μ-invariant equal to 0. The final section illustrates the general theory by a detailed discussion of the three Elliptic curves of conductor 11, at the prime ρ=5.
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