TITLE: Ranicki symmetric L-groups of finite fields?
QUESTION [10 upvotes]: 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!

REPLY [10 votes]: 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).