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Universal forms over $$\mathbb Q(\sqrt{5})$$. (English) Zbl 1178.11035
In this work, the author considers the universal forms over $$\mathbb{Q}(\sqrt{5})$$ (a form is called universal if it represents all positive integers). He derives all quaternary positive definite integral quadratic forms over $$\mathbb{Q}(\sqrt{5})$$ and also gives a proof of Conway and Schneeberger’s 15-Theorem. He lists in Theorem 3.3 that there are 35 quaternary integral quadratic forms over $$\mathbb{Q}(\sqrt{5})$$ which are universal. Also he shows in Corollary 3.4 that there are 58 nonisometric quaternary integral universal quadratic forms over $$\mathbb{Q}(\sqrt{5})$$. Finally, he proves that any $$O-$$lattice which represents $[1,2,1+\varepsilon ^{2},2+\varepsilon ^{\pm 2},2(1+\varepsilon ^{2}),3(1+\varepsilon ^{2})]$ is universal.

##### MSC:
 1.1e+11 Forms over real fields 1.1e+42 Class numbers of quadratic and Hermitian forms 1.1e+26 Sums of squares and representations by other particular quadratic forms
##### Keywords:
quadratic forms; universal form; class number
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##### References:
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