Let $Q\_1, Q\_2, R$ be quadratic froms over $\mathbb{Z}$ such that $Q\_1 \oplus R \cong Q\_2 \oplus R$ as quadratic forms. Is it necessary that $Q\_1 \cong Q\_2$?


I know that by Witt's theorem it is true for fields.



Are stably equivalent quadratic forms over Z equivalent?
I would say no; an example is when $R$ is a hyperbolic plane, the relation you demand says merely that $Q\_1$ and $Q\_2$ are in the same genus. This is in SPLAG, page 378 in the first (1988) edition, see also [Clark Jagy](http://alpha.math.uga.edu/~pete/Clark_Jagy_11_13_2013.pdf).


Notice that rational equivalence "without essential denominator" is exactly how Siegel defined the genus. So, $x^2 + 14 y^2$ and $2 x^2 + 7 y^2$ are rationally equivalent, and we can arrange that the denominators are prime to $14,$ but **not integrally equivalent**.


Give me a few minutes, I am going to display integer equivalence for $Q\_1 = x^2 + 14 y^2,$ $Q\_2 = 2 x^2 + 7 y^2,$ which pair take up a good deal of room in Cox's book, and $R = 2 z w.$ By solving $P^T A\_1P = A\_2$ in 4 by 4 integer matrices...


$$
A\_1 =
\left(
\begin{array}{rrrr}
 1 & 0 & 0 & 0 \\
 0 & 14 & 0 & 0 \\
 0 & 0 & 0 & 1 \\
 0 & 0 & 1 & 0
\end{array}
\right)
$$


$$
P =
\left(
\begin{array}{rrrr}
 2 & 7 & -2 & 2 \\
 1 & 5 & -2 & 1 \\
 -4 & -14 & 6 & -3 \\
 2 & 14 & -5 & 3
\end{array}
\right)
$$


$$
A\_2 =
\left(
\begin{array}{rrrr}
 2 & 0 & 0 & 0 \\
 0 & 7 & 0 & 0 \\
 0 & 0 & 0 & 1 \\
 0 & 0 & 1 & 0
\end{array}
\right)
$$


$$
\small
\left(
\begin{array}{rrrr}
 2 & 1 & -4 & 2 \\
 7 & 5 & -14 & 14 \\
 -2 & -2 & 6 & -5 \\
 2 & 1 & -3 & 3
\end{array}
\right)
\left(
\begin{array}{rrrr}
 1 & 0 & 0 & 0 \\
 0 & 14 & 0 & 0 \\
 0 & 0 & 0 & 1 \\
 0 & 0 & 1 & 0
\end{array}
\right)
\left(
\begin{array}{rrrr}
 2 & 7 & -2 & 2 \\
 1 & 5 & -2 & 1 \\
 -4 & -14 & 6 & -3 \\
 2 & 14 & -5 & 3
\end{array}
\right) =
\left(
\begin{array}{rrrr}
 2 & 0 & 0 & 0 \\
 0 & 7 & 0 & 0 \\
 0 & 0 & 0 & 1 \\
 0 & 0 & 1 & 0
\end{array}
\right)
$$


check with gp-pari


=====================



```
parisize = 4000000, primelimit = 500509
? a1 = [ 1,0,0,0; 0,14,0,0; 0,0,0,1; 0,0,1,0]
%1 = 
[1 0 0 0]

[0 14 0 0]

[0 0 0 1]

[0 0 1 0]

? a2 = [ 2,0,0,0; 0,7,0,0; 0,0,0,1; 0,0,1,0]
%2 = 
[2 0 0 0]

[0 7 0 0]

[0 0 0 1]

[0 0 1 0]

? p = [ 2,7,-2,2; 1,5,-2,1; -4,-14,6,-3; 2,14,-5,3]
%3 = 
[2 7 -2 2]

[1 5 -2 1]

[-4 -14 6 -3]

[2 14 -5 3]

? matdet(p)
%4 = 1
? pt = mattranspose(p)
%5 = 
[2 1 -4 2]

[7 5 -14 14]

[-2 -2 6 -5]

[2 1 -3 3]

? a1
%6 = 
[1 0 0 0]

[0 14 0 0]

[0 0 0 1]

[0 0 1 0]

? pt * a1 * p
%7 = 
[2 0 0 0]

[0 7 0 0]

[0 0 0 1]

[0 0 1 0]

? a2
%8 = 
[2 0 0 0]

[0 7 0 0]

[0 0 0 1]

[0 0 1 0]

? pt * a1 * p - a2
%9 = 
[0 0 0 0]

[0 0 0 0]

[0 0 0 0]

[0 0 0 0]

? 

```

===================


