Universal quadratic forms over multiquadratic fields

Abstract

For all positive integers $k$ and $N$ we prove that there are infinitely many totally real multiquadratic fields $K$ of degree $2^k$ over $\mathbb{Q}$ such that each universal quadratic form over $K$ has at least $N$ variables.

6 pages

References in corpus (3)

• Number fields without n-ary universal quadratic forms
• Universal quadratic forms and elements of small norm in real quadratic fields
• Norms of indecomposable integers in real quadratic fields

Cited by in corpus (12)

• Lifting problem for universal quadratic forms
• Universal quadratic forms, small norms and traces in families of number fields
• There are no universal ternary quadratic forms over biquadratic fields
• Number fields without universal quadratic forms of small rank exist in most degrees
• Pythagoras numbers of orders in biquadratic 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
• Lifting problem for universal quadratic forms over totally real cubic number fields
• Additive structure of non-monogenic simplest cubic fields
• Arithmetic of cubic number fields: Jacobi-Perron, Pythagoras, and indecomposables
• Quaternions and universal quadratic forms over number fields
• Generalizing Hurwitz's quaternionic proof of Lagrange's and Jacobi's four-square theorems