Lifting problem for universal quadratic forms

Abstract

We study totally real number fields that admit a universal quadratic form whose coefficients are rational integers. We show that $\mathbb{Q}(\sqrt{5})$ is the only such real quadratic field, and that among fields of degrees 3, 4, 5, and 7 which have principal codifferent ideal, the only one is $\mathbb{Q}({\zeta }_7+{\zeta }_7^{-1})$, over which the form $x^2+y^2+z^2+w^2+xy+xz+xw$ is universal. Moreover, we prove an upper bound for Pythagoras numbers of orders in number fields that depends only on the degree of the number field.

arXiv comment16 pages, incorporated referee comments

References in corpus (9)

• A database of number fields
• Number fields without n-ary universal quadratic forms
• Universal quadratic forms and elements of small norm in real quadratic fields
• Universal quadratic forms, small norms and traces in families of number fields
• There are no universal ternary quadratic forms over biquadratic fields
• Universal quadratic forms over multiquadratic fields
• Norms of indecomposable integers in real quadratic fields
• Indecomposable integers in real quadratic fields
• Enumeration of totally real number fields of bounded root discriminant

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• A cubic ring of integers with the smallest Pythagoras number
• On the Pythagoras number of the simplest cubic fields
• Minimal rank of universal lattices and number of indecomposable elements in real multiquadratic fields
• Universal quadratic forms and indecomposables in number fields: A survey
• Trace and norm of indecomposable integers in cubic orders
• On indefinite $k$-universal integral quadratic forms over number fields
• Lifting problem for universal quadratic forms over totally real cubic number fields
• On quadratic Waring's problem in totally real number fields
• Periodicity of general multidimensional continued fractions using repetend matrix form
• Universal quadratic forms and Northcott property of infinite number fields
• Sums of squares in $S$-integers
• On classic $n$-universal quadratic forms over dyadic local fields
• Additive structure of non-monogenic simplest cubic fields
• Sails for universal quadratic forms
• Arithmetic of cubic number fields: Jacobi-Perron, Pythagoras, and indecomposables
• Quaternions and universal quadratic forms over number fields
• Universal quadratic forms and Dedekind zeta functions
• Failures of integral Springer's Theorem
• Generalizing Hurwitz's quaternionic proof of Lagrange's and Jacobi's four-square theorems
• Pythagoras numbers for infinite algebraic fields