An invariant for quadratic forms valued in Galois Rings of characteristic 4

We introduce an invariant for nonsingular quadratic forms that take values in a Galois Ring of characteristic 4. This notion extends the invariant in ${\mathbb{Z}}_8$ for ${\mathbb{Z}}_4$-valued quadratic forms defined by Brown [E.H. Brown, Generalizations of the Kervaire invariant, Ann. of Math. (2) 95 (2) (1972) 368–383] and studied by Wood [J.A. Wood, Witt's extension theorem for mod four valued quadratic forms, Trans. Amer. Math. Soc. 336 (1) (1993) 445–461]. It is defined in the associated Galois Ring of characteristic 8. Nonsingular quadratic forms are characterized by their invariant and the type of the associated bilinear form (alternating or not).