$p$-retract Rationality and Norm One Tori

TL;DR

The paper investigates when the norm one torus $R^{(1)}_{K/k}{\mathbb G}_m$ attached to a non-Galois finite separable extension $K/k$ is $p$-retract rational over $k$, focusing on Galois closures with ${\rm Gal}(L/k)\cong S_n$ or $A_n$. It translates the problem into lattice theory via the flasque class $\rho_G(\hat{T})$ and the module $J_{G/H}$, showing that $p$-retract rationality corresponds to the $p$-invertibility of $\rho_G(J_{G/H})$. For the cases ${\rm Gal}(L/k)=S_n$, ${\rm Gal}(L/K)=S_{n-1}$ and ${\rm Gal}(L/k)=A_n$, ${\rm Gal}(L/K)=A_{n-1}$, the authors prove that $R^{(1)}_{K/k}{\mathbb G}_m$ is $p$-retract rational over $k$ precisely when $n$ is prime or $p$ is coprime to $n$. The proofs combine explicit group-theoretic constructions with known Galois-case criteria, yielding a clear, testable criterion for non-Galois extensions and extending rationality insights for algebraic tori.

Abstract

We study whether the norm one torus associated with a finite separable non-Galois field extension $K/k$ is $p$-retract rational over $k$ for a prime $p$, focusing on the case where the Galois group of the Galois closure of $K/k$ is either the symmetric or the alternating group.

Theorems & Definitions (29)

• Theorem 1.1
• Remark 1.2
• Theorem 2.1
• Theorem 2.2: EM75*Theorem 1.5 Salt84*Theorem 3.14
• Proposition 2.3: Endo11*Proposition 1.7
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Proposition 3.3