Let ( E )$E$ be an unramified quadratic extension of a local field ( F )$F$, with ( p )$p$ odd. Let ( E^1 )$E^1$ denote the set of norm 1$1$ elements of ( E )$E$. What can be said about the following index:
- ( [ (E^1 \cap (1 + P_E)) : E^1 \cap (1 + P_E^M)] )
where M $$ {\rm 1.}\ \ \ \ [ (E^1 \cap (1 + P_E )) : E^1 \cap (1 + P_E^M )] $$ where $M$ is a positive integer.
We know that in this case N_{E/F}(1+P_E)=1+P_F$N_{E/F}(1+P_E )=1+P_F$. If If we consider the short exact sequence
1--->E^1--->R_{E}^{\times}--->R_{F}^{\times}--->1
and $$ 1\longrightarrow E^1\longrightarrow R_{E}^{\times}\longrightarrow R_{F}^{\times}\longrightarrow 1 $$ and intersect this with 1+P_{E}$1+P_{E}$, we get
1--->E^1\cap (1+P_E)--->1+P_E---> 1+P_{F}--->1 $$ 1 \longrightarrow E^1\cap (1+P_E)\longrightarrow 1+P_E\longrightarrow 1+P_{F}\longrightarrow $$
but I don't think this helps in computing the index in 1.
