theorem A.340 Cartan's criterion for solvability

open in the book · appendices/A-long-proofs.tex:16561 · p. 2956

Rests on

depends_on lemma A.339 Trace lemma ¶
- depends_on lemma A.336 The adjoint of a Jordan decomposition ¶
  - depends_on theorem A.335 Jordan decomposition ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
    - proves proof app:A-long-proofs@proof-203 ¶
  - proves proof app:A-long-proofs@proof-204 ¶
- depends_on theorem A.335 Jordan decomposition ¶ ↺
- proves proof app:A-long-proofs@proof-207 ¶
depends_on theorem A.338 Engel ¶
- depends_on definition A.331 Derived series, solvable, ideal ¶
- depends_on lemma A.337 A nilpotent map has nilpotent adjoint ¶
  - depends_on definition 5.37 Linear transformation ¶ ↺
  - proves proof app:A-long-proofs@proof-205 ¶
- proves proof app:A-long-proofs@proof-206 ¶
proves proof app:A-long-proofs@proof-208 ¶

Supports

depends_on corollary A.342 Orthogonal splitting of ideals ¶
- depends_on corollary A.343 The Killing form of a simple ideal ¶
  - depends_on lemma A.409 A Casimir invertible on a nontrivial irreducible module ¶
    - depends_on theorem A.410 Whitehead's first and second lemmas ¶
      - depends_on corollary A.411 Semisimple algebras admit no nontrivial extension ¶
- depends_on lemma A.406 Splitting a submodule of codimension one ¶
  - depends_on theorem A.407 Weyl's complete reducibility theorem ¶
    - depends_on proposition A.408 The two groups vanish for the trivial module ¶
    - depends_on theorem A.410 Whitehead's first and second lemmas ¶ ↺
- depends_on lemma A.409 A Casimir invertible on a nontrivial irreducible module ¶ ↺
- depends_on proposition A.408 The two groups vanish for the trivial module ¶ ↺
- depends_on theorem A.407 Weyl's complete reducibility theorem ¶ ↺
depends_on lemma A.402 The trace form is nondegenerate for a faithful representation ¶
- depends_on lemma A.406 Splitting a submodule of codimension one ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Trace lemma	declared	appendices/A-long-proofs.tex:16572
`depends_on`	→	Engel	declared	appendices/A-long-proofs.tex:16572
`depends_on`	←	Orthogonal splitting of ideals	declared	appendices/A-long-proofs.tex:16693
`depends_on`	←	The trace form is nondegenerate for a faithful representation	declared	appendices/A-long-proofs.tex:19602
`proves`	←	app:A-long-proofs@proof-208	declared	appendices/A-long-proofs.tex:16575