LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Maximal residue-injective subfields exist

Statement

Assume the Axiom of Choice.

Let $(A, m)$ be an equicharacteristic local ring. Then there exists a subfield $K \subseteq A$ that is maximal, under inclusion, among subfields whose residue map to $A / m$ is injective.

Facts & Assumptions

Given: An equicharacteristic local ring $(A, m)$ and the Axiom of Choice.

[L1]

The residue field's prime field embeds in $A$ , so the family of residue-injective subfields is nonempty (The prime field lifts in the equicharacteristic case).

[L2]

Assuming the Axiom of Choice, every nonempty poset in which every chain has an upper bound has a maximal element (Zorn's lemma).

Proof

technique · apply Zorn's lemma to the poset of residue-injective subfields

1.1

Let $S$ be the set of subfields $K \subseteq A$ for which the residue map $K \to A / m$ is injective. By [L1], $S$ is nonempty. Order $S$ by inclusion.

L1givenconstruct

2.1

If $C \subseteq S$ is a chain, then $⋃_{K \in C} K$ is again a subfield of $A$ : closure under the field operations is inherited from some chain member containing the finitely many elements involved. Its residue map is still injective, because a nonzero element of the union already lies in one chain member where injectivity holds. Thus every chain in $S$ has an upper bound in $S$ .

step 1.1givenalgebra

3.1

By [L2], the poset $S$ has a maximal element. That is exactly a maximal residue-injective subfield of $A$ .

L2step 2.1∎

Depends on

Used by

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Melvin Hochster, The structure theory of complete local rings (standard reference, not scraped)