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LSpice
LSpice
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The existence of integral quadratic forms with prescribed “local” data (i.e., essentially the genus) has been also proved by Nikulin. For a proof of Nikulin's existence theorem, see for example http://www.math.ucsb.edu/~drm/manuscripts/eiqf.pdfMiranda and Morrison - Embeddings of Integral Quadratic Forms, Theorem $5.2$5.2.

On the other hand, Conway and Sloane do provide some references -– perhaps they are helpful. They write at the beginning of section $7$7: "No proofs will be offered. For $p\neq 2$ several accounts are readily available (e.g. Cassels, [Cas3]), and all cases are handled by O'Meara [O'Me 1], ...… etc.".

The existence of integral quadratic forms with prescribed “local” data (i.e., essentially the genus) has been also proved by Nikulin. For a proof of Nikulin's existence theorem, see for example http://www.math.ucsb.edu/~drm/manuscripts/eiqf.pdf, Theorem $5.2$.

On the other hand, Conway and Sloane do provide some references - perhaps they are helpful. They write at the beginning of section $7$: "No proofs will be offered. For $p\neq 2$ several accounts are readily available (e.g. Cassels, [Cas3]), and all cases are handled by O'Meara [O'Me 1], ... etc.".

The existence of integral quadratic forms with prescribed “local” data (i.e., essentially the genus) has been also proved by Nikulin. For a proof of Nikulin's existence theorem, see for example Miranda and Morrison - Embeddings of Integral Quadratic Forms, Theorem 5.2.

On the other hand, Conway and Sloane do provide some references – perhaps they are helpful. They write at the beginning of section 7: "No proofs will be offered. For $p\neq 2$ several accounts are readily available (e.g. Cassels, [Cas3]), and all cases are handled by O'Meara [O'Me 1], … etc.".

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Dietrich Burde
Dietrich Burde
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The existence of integral quadratic forms with prescribed “local” data (i.e., essentially the genus) has been also proved by Nikulin. For a proof of Nikulin's existence theorem, see for example http://www.math.ucsb.edu/~drm/manuscripts/eiqf.pdf, Theorem $5.2$.

On the other hand, Conway and Sloane do provide some references - perhaps they are helpful. They write at the beginning of section $7$: "No proofs will be offered. For $p\neq 2$ several accounts are readily available (e.g. Cassels, [Cas3]), and all cases are handled by O'Meara [O'Me 1], ... etc.".