On the fourth power level of $\mathfrak{p}$-adic completions of biquadratic number fields

TL;DR

This work investigates the fourth level $s_4(K)$ of quartic/biquadratic number fields by analyzing their $\mathfrak{p}$-adic completions at primes above $2$. It establishes a finite, tau-based decision procedure for $s_4(K_\mathfrak{p})$ in the main case $e=4$, $f=1$ using congruence criteria modulo $\mathfrak{p}^{13}$ derived via Hensel lifting. The method is then specialized to biquadratic fields $K=\mathbb{Q}(\sqrt{m},\sqrt{n})$, with Macaulay2-based computations yielding that $s_4(K_\mathfrak{p})$ lies in $\{1,2,3,4,6,15\}$ for primes $\mathfrak{p}\mid(2)$, under various congruence conditions on $m,n,k$. The results motivate a conjecture that the global fourth level equals the maximum of the local ones, and they highlight open questions about the full set of possible $s_4(K)$ values across number fields.

Abstract

Let $K$ be a number field and $\mathfrak{p} \mid (2)$ be a prime ideal. We compute the fourth level of the $\mathfrak{p}$-adic completions of $K$ when the ramification index is $4$ and the inertial degree is trivial for the ideal $\mathfrak{p}$. This enables the computation of the fourth level of any $\mathfrak{p}$-adic completion of any quartic number field. Here we apply this result to biquadratic number fields and obtain lower bounds for the fourth level of such number fields.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof