Recently I learned here interesting facts about $\mathbb F_p((t))$, the field of the formal Laurent series with coefficients in $\mathbb F_p$. I looked up quite a few papers on local fields but I could not seem to find an explicit formula for an involution (for example a field automorphism of order 2) on $\mathbb F_p((t))$.
Involutions on $\mathbb F_p((t))$ are interesting for me because I want to investigate Cayley-Dickson Algebra over $\mathbb F_p((t))$. To do s, I use the Cayley Dickson construction as in here which I will cite for convenience:
"The Cayley-Dickson construction. The main tool for dealing with conic algebras in general and composition algebras in particular is the Cayley-Dickson construction. Its inputs are a conic algebra $B$ and a non-zero scalar $\mu \in k$. Its output is a conic algebra $C:=Cay(B,\mu)$ that is given on the vector space direct sum $C= B \oplus Bj$ of two copies of $B$ by the multiplication.
(1) $(u_1+v_1j)(u_2+v_2j)=(u_1u_2+\mu v_2^*v_1)+(v_2u_1+v_1u_2^*)j$
(...)"
To check if $\mathbb F_p((t))$ is what Petersson calls a Conic Algebra (other authors apparently use the term quadratic algebras or algebras of degree 2) I need to find an involution $\varphi: \mathbb F_p((t)) \rightarrow \mathbb F_p((t)), x\mapsto \varphi(x)$ such that $\varphi(x)x=|x|_t$ where $|x|_t$ denotes the t-adic norm on $\mathbb F_p((t))$. I tried to use $x^{-1}|x|_t$. It works just fine for an Element of the form $x=\sum \limits_{i=z} ^\infty a_it^i$ with $z\in \mathbb Z$ and $a_z\neq 0$ since $x^{-1}|x|_t$ turns out to be $\frac{|x|_t^2}{\sum \limits _{i=0} a_{i+z}t^i}$. Does it also work with elements of the form $x=\sum \limits_{i=-\infty} ^\infty a_it^i$ or is there a different involution?
sincerely slinshady
There are no elements in $\Bbb F_p((t))$ with infinitely many negative powers of $t$; all of them are of the first form you described. (Indeed there can be no multiplicative structure on the set of infinite sums of the second form you described, since multiplication of them is ill-defined in general.)