Corollary

The hypotheses and notation being those of prop. 3, the elements $a.T_1$ ($a \in A^*$) form a set of generators of the group of types of quadratic forms over A.

To seek the structure of the group of types of quadratic forms over A is therefore to seek the $\mathbf{Z}$-linear relations which exist between the elements of the form $a.T_1$. If $b \in A^*$, the form $Q_1$ defined in prop. 4 is manifestly equivalent to $b^2Q_1$; hence $a.T_1 = ab^2.T_1$, which shows that $a.T_1$ depends only on the class of $a$ modulo the subgroup $(A^*)^2$ of squares of elements of $A^*$. Moreover, it follows from prop. 3 that we have $(-a).T_1 = -a.T_1$. However, in general there are other $\mathbf{Z}$-linear relations between the $a.T_1$ than those which follow from the relations we have just indicated.