I read this article "Local class field theory via Lubin-Tate theory" http://arxiv.org/pdf/math/0606108v2.pdf. And I want to find the maximal abelian extensions for quadratic extensions of $\mathbb Q_p,\ p>2$. Of course there are only 3 quadratic extensions: $\mathbb Q_p(\sqrt{p}), \mathbb Q_p(\sqrt{p\varepsilon}),\mathbb Q_p(\sqrt{p\varepsilon})$(let $K$ be one of this extensions, and $\mathbb F_q$ its residue field), where $\varepsilon$ isn't quadratic residue in $\mathbb F_p$. According to this theory I just need to find a polynomial $f \in \mathcal O_K[T]$, such that $$ f = \pi T\mod deg \ 2, f=T^q \mod \mathfrak m , $$ such that f has "good roots", and $f_m=f^{\phi^{m-1}} \circ ... f^{\phi} \circ f$ too. In the case of $\mathbb Q_p$ there is "good polynomial" $f=(T+1)^p-1$. But I can't create a "good polynomial" for quadratic case. Is there any good polynomials for quadratic cases?

P.S. I have heard something that formal groups are related to elliptic curves, but I don't know much about latter. I'll be glad for any references to this theme.

I read this article "Local class field theory via Lubin-Tate theory" http://arxiv.org/pdf/math/0606108v2.pdf. And I want to find the maximal abelian extensions for quadratic extensions of $\mathbb Q_p,\ p>2$. Of course there are only 3 quadratic extensions: $\mathbb Q_p(\sqrt{p}), \mathbb Q_p(\sqrt{\varepsilon}),\mathbb Q_p(\sqrt{p\varepsilon})$ (where $\varepsilon$ is not a quadratic residue in $\mathbb F_p$). Let $K$ be one of these extensions, and $\mathbb F_q$ its residue field. According to this theory I just need to find a polynomial $f \in \mathcal O_K[T]$, such that $$ f = \pi T\mod deg \ 2, f=T^q \mod \mathfrak m , $$ such that f has "good roots", and $f_m=f^{\phi^{m-1}} \circ ... f^{\phi} \circ f$ too. In the case of $\mathbb Q_p$ there is "good polynomial" $f=(T+1)^p-1$. But I can't create a "good polynomial" for quadratic case. Is there any good polynomials for quadratic cases?

P.S. I have heard something that formal groups are related to elliptic curves, but I don't know much about latter. I'll be glad for any references to this theme.

I read the this article "Local class field theory via Lubin-Tate theory" http://arxiv.org/pdf/math/0606108v2.pdf. And I want to find the maximal abelian extensions for quadratic extensions of $\mathbb Q_p,\ p>2$. Of course there are only 3 quadratic extensions: $\mathbb Q_p(\sqrt{p}), \mathbb Q_p(\sqrt{p\varepsilon}),\mathbb Q_p(\sqrt{p\varepsilon})$(let $K$ be one of this extensions, and $\mathbb F_q$ its residue field), where $\varepsilon$ isn't quadratic residue in $\mathbb F_p$. According to this theory I just need to find a polynomial $f \in \mathcal O_K[T]$, such that $$ f = \pi T\mod deg \ 2, f=T^q \mod \mathfrak m , $$ such that f has "good roots", and $f_m=f^{\phi^{m-1}} \circ ... f^{\phi} \circ f$ too. In the case of $\mathbb Q_p$ there is "good polynomial" $f=(T+1)^p-1$. But I can't create a "good polynomial" for quadratic case. Is there any good polynomials for quadratic cases?

P.S. I have heard something that formal groups are related to elliptic curves, but I don't know much about latter. I'll be glad for any references to this theme.

I read this article "Local class field theory via Lubin-Tate theory" http://arxiv.org/pdf/math/0606108v2.pdf. And I want to find the maximal abelian extensions for quadratic extensions of $\mathbb Q_p,\ p>2$. Of course there are only 3 quadratic extensions: $\mathbb Q_p(\sqrt{p}), \mathbb Q_p(\sqrt{p\varepsilon}),\mathbb Q_p(\sqrt{p\varepsilon})$(let $K$ be one of this extensions, and $\mathbb F_q$ its residue field), where $\varepsilon$ isn't quadratic residue in $\mathbb F_p$. According to this theory I just need to find a polynomial $f \in \mathcal O_K[T]$, such that $$ f = \pi T\mod deg \ 2, f=T^q \mod \mathfrak m , $$ such that f has "good roots", and $f_m=f^{\phi^{m-1}} \circ ... f^{\phi} \circ f$ too. In the case of $\mathbb Q_p$ there is "good polynomial" $f=(T+1)^p-1$. But I can't create a "good polynomial" for quadratic case. Is there any good polynomials for quadratic cases?

P.S. I have heard something that formal groups are related to elliptic curves, but I don't know much about latter. I'll be glad for any references to this theme.