Hello , Let f be a quadratic form of rank n over $\mathbb{Z}/2\mathbb{Z}$. Then:

(i) If n is odd, we have

$f \simeq x_2 + (b_1 {y_1}^2 + {y_1} {z_1} + d_1 {z_1}^2)+(b_2 {y_2}^2 + {y_2} {z_2} + d_2 {z_2}^2)+ \cdots +(b_l {y_l}^2 + {y_l} {z_l} + d_l {z_l}^2)$

(ii) If n is even, we have $f \simeq (b_1 {y_1}^2 + {y_1} {z_1} + d_1 {z_1}^2)+(b_2 {y_2}^2 + {y_2} {z_2} + d_2 {z_2}^2)+ \cdots +(b_l {y_l}^2 + {y_l} {z_l} + d_l {z_l}^2)$

for some $l \in \mathbb{N} , b_l,d_l \in \mathbb{Z}/2\mathbb{Z}$

Question : is this theorem valid for quadratic forms over $\mathbb{Z}/{2^r}\mathbb{Z} ,r>1$.?

if not , what one have in this case ?

Best regards.

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Get Rational Quadratic Forms by J.W.S. Cassels. Canonical forms in $\mathbb Z_2$ are discussed on pages 117-118, giving some (not necessarily exclusive though). Then, at the top of page 120, mention is made of articles that complete the job, B. W. Jones 1944, Gordon Pall 1945, finally G. L. Watson 1976. –  Will Jagy Feb 17 at 18:48