The Homogeneous Spectrum of Milnor-Witt $K$-Theory

For any field $F$ (of characteristic not equal to 2), we determine the Zariski spectrum of homogeneous prime ideals in $K_{\ast }^{MW}(F)$, the Milnor-Witt $K$-theory ring of $F$. As a corollary, we recover Lorenz and Leicht’s classical result on prime ideals in the Witt ring of $F$. Our computation can be seen as a first step in Balmer’s program for studying the tensor triangular geometry of the stable motivic homotopy category.