Explicit descriptions of the subfields $(NL)^{pi}$ and $(NL)^{pi}(NL)^{sep}$ of $NL$ and new explicit criteria for $NL = (NL)^{pi}(NL)^{sep}$

Technical Summary of "Explicit descriptions of the subfields (NL)pi and (NL)pi(NL)sep of NL and new explicit criteria for NL = (NL)pi(NL)sep"

Problem Statement
The paper addresses the structural analysis of finite field extensions in prime characteristic $p > 0$. Specifically, it focuses on simple finite field extensions $L/K$ where $L = K(\theta) \cong K[x]/(f(x))$, and the compositum $NL$ formed by $L$ and a purely inseparable extension $N/K$. The central problem is to provide explicit, constructive descriptions of the maximal purely inseparable subfield ($L_{pi}$), the maximal separable subfield ($L_{sep}$), and their compositum ($L_{pi}L_{sep}$) within $L$. Furthermore, the paper investigates the analogous subfields within the extension $NL/N$, denoted as $(NL)_{pi}$ and $(NL)_{sep}$.

A key issue in this domain is that, in general, the compositum $L_{pi}L_{sep}$ is a proper subfield of $L$ (i.e., $L_{pi}L_{sep} \subsetneq L$). The paper seeks to quantify exactly how far $L$ deviates from being the tensor product of its separable and purely inseparable parts by determining the degrees of these extensions and the specific generators of these subfields in terms of the coefficients of the minimal polynomial $f(x)$.

Methodology
The author employs a constructive algebraic approach based on the "separable presentation" of polynomials. For a minimal polynomial $f(x) \in K[x]$, the paper utilizes the unique decomposition $f(x) = f_{sep}(x^{p^n})$, where $f_{sep}$ is separable and $n$ is the inseparability degree.

The methodology relies on:

1. Field Invariants: The definition and utilization of numerical invariants, specifically $m_f$ (for $L/K$) and $m_{f,N}$ and $m_{f,N}(\theta)$ (for $NL/N$). These invariants are defined as the maximal integers $m'$ such that specific $p^{m'}$-th roots of the coefficients of $f(x)$ lie within the relevant field extensions.
2. Explicit Construction: Deriving the generators of the subfields $L_{pi}$, $L_{sep}$, and their compositum directly from the coefficients $\lambda_i$ of the polynomial $f(x) = \sum \lambda_i x^{ip^n}$.
3. Degree Calculations: Computing explicit numerical values for degrees such as $[L_{pi}:K]$, $[L_{sep}:K]$, $[L_{pi}L_{sep}:L_{pi}]$, and the defect degree $[L : L_{pi} \otimes L_{sep}]$.
4. Compositum Analysis: Extending these results to the case where $N/K$ is a purely inseparable extension, analyzing the structure of $NL = N(\theta)$ and determining how the purely inseparable and separable parts of $NL$ relate to $N$ and the original coefficients.

Key Contributions and Results

1. Explicit Descriptions for Simple Extensions $L/K$
The paper provides a complete characterization of the subfields of a simple extension $L = K(\theta)$ with minimal polynomial $f(x) = \sum_{i=0}^s \lambda_i x^{ip^n}$:

• Maximal Separable Subfield: $L_{sep} = K(\theta^{p^n})$, with degree $[L_{sep}:K] = s = \deg(f_{sep})$.
• Maximal Purely Inseparable Subfield: $L_{pi} = K(\lambda_0^{1/p^m}, \dots, \lambda_{s-1}^{1/p^m})$, where $m = m_f$ is the invariant defined by the maximal power of $p$ for which the coefficients' roots lie in $L_{pi}$. The degree is $[L_{pi}:K] = p^m$.
• The Compositum: $L_{pi}L_{sep} = L_{pi}(\theta^{p^{n-m}})$. The paper proves that $L_{pi}L_{sep} \cong L_{pi} \otimes_K L_{sep}$.
• The Defect: The extension $L$ over $L_{pi}L_{sep}$ is a simple purely inseparable extension of exponent $n-m$, with degree $p^{n-m}$. This explicitly quantifies why $L_{pi}L_{sep}$ is often a proper subfield of $L$.

2. Explicit Descriptions for Compositum Extensions $NL/N$
For $L/K$ simple and $N/K$ purely inseparable, the paper describes the subfields of $NL/N$:

• It introduces invariants $m_{f,N}$ and $m_{f,N}(\theta)$ associated with the extension $NL/N$.
• Structure of $NL$: The minimal polynomial of $\theta$ over $N$ is explicitly given as $f_N(x) = \sum \lambda_i^{1/p^{m_{f,N}}} x^{i p^{n-m_{f,N}}}$.
• Subfields:
  • $(NL)_{sep} = N(\theta^{p^{n-m_{f,N}}})$.
  • $(NL)_{pi} = N(\lambda_0^{1/p^{m_{f,N} + m_{f,N}(\theta)}}, \dots, \lambda_{s-1}^{1/p^{m_{f,N} + m_{f,N}(\theta)}})$.
  • The compositum $(NL)_{pi}(NL)_{sep}$ is described similarly to the $L/K$ case, with degrees determined by the sum of the invariants.
• The paper clarifies that $(NL)_{pi}$ generally strictly contains $NL_{pi}$ (the purely inseparable part of $L$ extended by $N$), providing the precise degree of this containment.

3. New Explicit Criteria for Equality $L = L_{pi}L_{sep}$
The paper establishes new, necessary, and sufficient conditions for the equality $L = L_{pi}L_{sep}$ (equivalently $L \cong L_{pi} \otimes L_{sep}$):

• For Simple Extensions: $L = L_{pi}L_{sep}$ if and only if $m_f = n$. This is equivalent to the condition that $\lambda_i^{1/p^n} \in L$ for all $i$, or that the coefficients of the $p^n$-th root of $f$ lie in the purely inseparable part of $L$.
• For General Finite Extensions: For a general finite extension $L = L_1 \cdots L_\nu$ (compositum of simple extensions), the paper provides a criterion based on the coefficients $\lambda_{ij}$ of the minimal polynomials of the generators. Specifically, $L = L_{pi}L_{sep}$ if and only if $\lambda_{ij}^{1/p^{n_i}} \in L_{pi}$ for all generators and coefficients.

Significance and Claims
The author claims that the primary significance of this work lies in moving beyond abstract existence theorems to explicit, computable descriptions.

• Explicitness: Unlike previous results that may describe these subfields abstractly, this paper provides formulas for the generators of $L_{pi}$ and $L_{sep}$ directly in terms of the coefficients of the defining polynomial $f(x)$.
• Quantification of the Defect: The paper explicitly calculates the degree $[L : L_{pi}L_{sep}] = p^{n-m_f}$, revealing the exact "gap" between the field and its separable-purely inseparable compositum.
• New Criteria: The paper offers new criteria for the equality $L = L_{pi}L_{sep}$ that are checkable via the coefficients of the minimal polynomial, extending known criteria (such as the degree criterion or separability over the purely inseparable part) into a more constructive framework.
• Generalization: The results generalize from simple extensions to the case where a purely inseparable extension $N$ is adjoined, providing a unified framework for analyzing the structure of $NL$.

The paper does not propose experimental applications or future implications beyond the theoretical advancement of field theory in positive characteristic. It positions itself as a foundational contribution to the structural theory of finite field extensions, providing tools for explicit calculation and classification.