2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/202838We present two compact derivations of the correct definition of the Chern-Simons term in the topologically non trivial context of thermal $QED_3$. One is based on a transgression descent from a D=4 background connection, the other on embedding the abelian model in SU(2). The results agree with earlier cohomology conclusions and can be also used to justify a recent simple heuristic approach. The correction to the naive Chern-Simons term, and its behavior under large gauge transformations are displayed.9 pages, RevTex, no figures, new derivation from non abelian embedding addedHigh Energy Physics - Theorytext