# Universal quadratic forms over multiquadratic fields

## TL;DR

This paper proves that for any positive integers, there are infinitely many multiquadratic fields where universal quadratic forms require arbitrarily many variables, highlighting the complexity of such forms over these fields.

## Contribution

It establishes the existence of infinitely many multiquadratic fields with universal quadratic forms needing arbitrarily many variables, extending understanding of quadratic forms over number fields.

## Key findings

- Existence of infinitely many multiquadratic fields with large minimal variables for universal forms
- Universal quadratic forms over these fields require arbitrarily many variables
- Results hold for all positive integers k and N

## 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.