Open Mathematics Courses

Invariants and finiteness

Binary covariants, averaging and Hilbert finiteness; polarization, vector invariants and symbolic evaluation; finite groups and Noether’s degree bounds; invariant rings of ternary cubics and quartics; finiteness in every characteristic and over Noetherian bases.

  1. Covariants, averaging and Hilbert finiteness
  2. Polarization, vector invariants and symbolic evaluation
  3. Invariants of finite groups and Noether's degree bound
  4. Invariant rings of ternary cubics and quartics
  5. Finiteness for finite groups in every characteristic

Written and self-checked by GPT-6.1 Sol (OpenAI) in Codex at Ultra. Original modern lessons, proofs, exercises and diagrams: CC0.

Editable sources and exact calculations

Download the course sources. Markdown, matching LaTeX, editable figures, calculation programs and exact quartic certificates. Build instructions.

Emmy Noether: historical readings

Noether’s original papers appear in German alongside extracts from the maintained English edition, 2026.08.13-r1. These historical PDF readings are separate from the modern lessons. German files reproduce the original journal pages; the English files are from the maintained translation. Original papers are public domain; the translation layer is CC0. Embedded fonts retain their actual terms.

  1. 1. On the formation of the system of ternary quartic forms — German PDF; English PDF. Sitzungsberichte der physikalisch-medizinischen Sozietät in Erlangen 39 (1907), 176–179
  2. 2. The system of ternary quartic forms — German PDF; English PDF. Journal für die reine und angewandte Mathematik 134 (1908), 23–90 and two tables
  3. 3. Invariant theory of forms in several variables — German PDF; English PDF. Jahresbericht der Deutschen Mathematiker-Vereinigung 19 (1910), 101–104
  4. 4. Invariant theory of forms in several variables — German PDF; English PDF. Journal für die reine und angewandte Mathematik 139 (1911), 118–154
  5. 7. Finiteness for invariants of finite groups — German PDF; English PDF. Mathematische Annalen 77 (1916), 89–92
  6. 8. Polynomial expressions for invariants of arbitrarily many forms — German PDF; English PDF. Mathematische Annalen 77 (1916), 93–102
  7. 15. Finiteness of integral invariants of binary forms — German PDF; English PDF. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen (1919), 138–156
  8. 16. Series expansion in form theory — German PDF; English PDF. Mathematische Annalen 81 (1920), 25–30
  9. 29. Finiteness in characteristic p — German PDF; English PDF. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen (1926), 28–35

English edition: Emmy Noether, collected papers in English.