I am working at a Problem for some time and it comes down to the question:

Let $K:=\mathbb{Q}_2[\zeta]$ be a cyclotomic extention of the dyadic field $\mathbb{Q}_2$. For any $c \in \mathbb{Q}_2[\zeta+\zeta^{-1}]$ one can define the trace form $b: K\times K \rightarrow \mathbb{Q}_2$, $(x,y) \mapsto trace (c x\overline{y})$. A standard computation shows that $b$ is a bilinear form and $(K,b)$ is a quadratic space. Lets further suppose the dimension of the $\mathbb{Q}_2$-Space $\mathbb{Q}_2[\zeta]$ is even.

My Problem is, that I can not figure out, wether this space is hyperbolic or not. One possible way to do it, is the computation of the Determinant, but I also don't know how to do that. Does anyone have an idea?

Laura Duebel

  • $\begingroup$ What is the automorphism $x \mapsto \overline{x}$? (it looks like complex conjugation, but that doesn't make sense $2$-adically.) Do you want an answer for one $c$ or for all $c$? Are there any restrictions on the order of $\zeta$? $\endgroup$ – Epargyreus Jun 4 '15 at 8:17
  • $\begingroup$ 1.The automorphism $x \mapsto \overline{x}$ is the conjugation $\zeta \mapsto \zeta^{-1}$. 2. Yes, I would like to know the answer for every $c$. 3. There is no restriction on the order of $\zeta$. $\endgroup$ – Laura Jun 4 '15 at 13:11
  • $\begingroup$ The map $\zeta \rightarrow \zeta^{-1}$ is not an automorphism in general. If $\zeta$ has order $2^m \cdot n$ for $n$ odd, then it comes from an automorphism if and only if there exists a $k$ such that $2^k \equiv -1 \pmod n$. Finally, can you clarify what you mean by hyperbolic? Do you want the form to be a direct sum of copies of the quadratic form $xy$ on $\mathbf{Q}^2_2$? $\endgroup$ – Epargyreus Jun 4 '15 at 23:07

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