Milnor-Witt K-theory for finite fields Are $K^{MW}_*(\mathbb{F_q})$ and $K^{MW}_n(\mathbb{F_q})$ already known? Where can I read about it?
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There is a short exact sequence for every $n$
$$ 0 \to I^{n - 1}(k) \to K_n^{MW}(k) \to K_n^M(k) \to 0 $$
This is proven independently of Morel's article by Gille, Scully, Zhong in "Milnor-Witt $K$-groups of local rings.", see Def.3.7, Thm.3.8, Thm.5.4.
The Milnor $K$-theory of a finite field appears in Milnor's original article "Algebraic $K$-theory of quadratic forms" as Example 1.5.
For the Witt groups and powers of augmentation ideals, see page 36, 37 and Theorem 3.5 in Lam's book "Introduction to quadratic forms over fields".
The culmination of these is:
$$ K^{MW}_{\geq 0}(\mathbf{F}_q) = (\mathbb{Z} \oplus \mathbb{Z} / 2) \oplus (\mathbb{F}_q^*) \oplus 0 \oplus 0 \dots$$
and for $n < 0$ we have 
$$K^{MW}_n(\mathbf{F}_q) = \mathbb{Z} / 4$$ if $q$ is 3 mod 4, and $$K^{MW}_n(\mathbf{F}_q) = \mathbb{Z} / 2[\epsilon] / \epsilon^2$$ if $q$ is 1 mod 4.
A: The Milnor-Witt K-theory for finite fields can be put together using 
$$
K^{MW}_n\cong K^M_n\times_{I^n/I^{n+1}}I^n
$$
using the known results for Milnor K-theory and the Witt ring. A treatment of the Witt ring dealing with all characteristics can be found in Elman-Karpenko-Merkurjev "The algebraic and geometric theory of quadratic forms". 
Addendum: the formula for Milnor-Witt K-theory can be found in Morel's paper "Sur les puissances de l'idéal fondamental de l'anneau de Witt".
