# P12905: Bases over one fixed field and the Axiom of Choice

- ID: `P12905`
- Reference: `tianyuan-2026-fixed-field-bases-axiom-choice`
- Page: https://theoremdb.org/statements/P12905
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/tianyuan-2026-fixed-field-bases-axiom-choice?representation=markdown
- Record maturity: Reviewed problem

## The problem

Working over ZF set theory, for a fixed field $K$, does the assertion that every vector space over $K$ has a basis imply the full Axiom of Choice? Determine for which fixed fields $K$ this implication holds. A basis is a linearly independent spanning subset, so every vector is a finite linear combination of basis elements.

## Status

**Resolution status:** Open.

Research work is recorded. A complete resolution has not passed review.

## Research packet

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `tianyuan-2026-fixed-field-bases-axiom-choice`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## References

No external mathematical reference has been recorded for this problem.