An elementary formulation of Kubota's proof of Satz 4 about biquadratic number fields

TL;DR

The paper presents an elementary reformulation of Kubota's Satz Vier for biquadratic number fields, enabling the computation of the Pólya group $|\mathcal P o(K)|$ from the Pólya groups of the three quadratic subfields and ramification data. It reframes Kubota's intricate proof by deriving an explicit description of the kernel and cokernel of the map $\psi$ that ties $\mathcal P o(k_1)\times\mathcal P o(k_2)\times\mathcal P o(k_3)$ to $\mathcal P o(K)$, using a sequence of lemmas on units and Hilfssätze to express $|\mathrm{Ker}\,\psi|$ in terms of the unit index $q_K$, ramification $s_K$, and invariants $\nu_K$, $i_2$, $j_2$. The main result is a closed formula for $|\mathcal P o(K)|$ that depends on whether $K$ is real or complex, and incorporates the interplay between the quadratic subfields' unit groups and the ramification data. This provides a practical tool for computing Polya groups in biquadratic fields and connects to the theory of strongly ambiguous classes and integer-valued polynomials.

Abstract

The aim of this note is to give an elementary formulation of Tomio Kubota's proof of Satz 4 in his paper in Nagoya Math. t. 10 (1956) since this proposition is a cornerstone for the computation of the order of the Polya group of the biquadratic number fields.

Theorems & Definitions (26)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1: Kubota's Hilfssatz 1
• proof
• Lemma 2.2
• proof
• Lemma 2.3: Kubota's Hilfssatz 2
• proof
• Lemma 2.4: Kubota's Hilfssätze 3 and 4
• proof