TITLE: Milnor-Witt K-theory for finite fields
QUESTION [5 upvotes]: Are $K^{MW}_*(\mathbb{F_q})$ and $K^{MW}_n(\mathbb{F_q})$ already known? Where can I read about it?

REPLY [8 votes]: I wanted to leave this as a comment, but I don't have enough reputation points yet.
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.