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I'm trying to figure out a way of understanding the kernel of the following map: Let $K$ be a quadratic field extension. Let V be the vector space over $\mathbb{F}_{2}$ of the ramified prime ideals of $K$. We define $F$ as the following map: $F$ gets $v\in V$ and maps it to its narrow ideal class of $K$.

What can we say about $F's$ kernel? It's non-trivial, but I cannot seem to find a way of describing it explicitly.

Thanks.

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What is a "narrow ideal class"? –  DaveUM Jun 21 '13 at 19:16
    
@DaveUM The narrow ideal class group of a field K is defined to be ${I_{K}} modulu {P_{K}^{+}}$, where ${I_{K}}$ is the group of fractional ideals of K and $P_{K}^{+}$ is the group of totally positive fractional ideal of K. –  Lenny Jun 23 '13 at 16:14

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