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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:
11E10 Forms over real fields
11E41 Class numbers of quadratic and Hermitian forms
11E25 Sums of squares and representations by other particular quadratic forms
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