Introduction to entanglement-assisted quantum error correction from pre-quantum mathematical literature

The presentation file consists of two parts. In Part 1: Overlooked incompatibility between finite fields and quantum error correction with its remedies, we give a simple counterexample to the popular definition of quantum stabilizer codes invoving finite fields, and review two mathematically correct ways to define them. In Part 2: Introduction to entanglement-assisted quantum error correction from pre-quantum mathematical literature, based on Part 1 we introduce entanglement-assisted quantum error correction by using as much finite fields and as little complex vectors/matrices as possible, which is hopefully more friendly to classical coding theorests. We also clarify the connection to pre-quantum mathematical literature. These slides will be used in a workshop held on November 2026 at Castelló University Institute of Mathematics and Applications (IMAC), Universitat Jaume I, Spain. Version 1 has redundant assumptions in the Witt extension theorem, which are removed in version 2. The other file consists of mathematical proofs omitted from the presentation. It is generated by AI.