Let ( E ) be an unramified quadratic extension of a local field ( F ), with ( p ) odd. Let ( E^1 ) denote the set of norm 1 elements of ( E ). What can be said about the following index:

1. ( [ (E^1 \cap (1 + P_E)) : E^1 \cap (1 + P_E^M)] )

where M is a positive integer.

We know that in this case N_{E/F}(1+P_E)=1+P_F. If we consider the short exact sequence

1--->E^1--->R_{E}^{\times}--->R_{F}^{\times}--->1

and intersect this with 1+P_{E}, we get

1--->E^1\cap (1+P_E)--->1+P_E---> 1+P_{F}--->1

but I don't think this helps in computing the index in 1.