Can anyone tell me what the Ranicki symmetric L-groups $L^\*(F)$ are when $F$ is a finite field? (and maybe provide a reference?) Thanks!



Ranicki symmetric L-groups of finite fields?
The symmetric $L$-group $L^\*(F)$ of a field $F$ are 4-periodic, 
$$L^n(F)=L^{n+4}(F)$$
by Proposition 7.1 of 
<http://www.maths.ed.ac.uk/~aar/papers/ats1.pdf>


$L^{2i}(F)$ is the Witt group of $(-)^i$-symmetric forms:
see Milnor and Husemoller!


$L^{2i+1}(F)=0$, see 
<http://www.maths.ed.ac.uk/~aar/papers/simple.pdf>
(my shortest paper).


