Captain: Henry Yuen
Fundamental Theorem of AlgebraTextbook
Show that every nonconstant complex polynomial has a complex root.
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Algebra
6 missions · 6 completed
The study of algebraic structures — groups, rings, and fields — and, through algebraic geometry, the geometry of the solution sets of polynomial equations. Using commutative algebra to describe these varieties, the field provides a common language of symmetry and structure that underlies much of modern mathematics.
Missions
Captain: tianyipeng
Hefferon Linear Algebra V: Jordan Canonical FormTextbook
Chapter Five of Jim Hefferon's *Linear Algebra* is one long search for a canonical form for matrix similarity, and Theorem IV.2.8 ends it: over the complex numbers every square matrix is similar to a matrix in Jordan form. That is the goal theorem of this mission and the capstone of the book. Mathlib carries the generalized eigenspace decomposition but has no Jordan canonical form, so this is a genuine target rather than a wrapper around an existing lemma; the Jordan block and the block-diagonal Jordan matrix are supplied as a mission definition. The milestones are the three results the proof is assembled from: diagonalizability as the existence of an eigenbasis, Cayley-Hamilton, and the canonical form of a nilpotent map, which is Jordan form applied to $t - \lambda$ on each generalized eigenspace.
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Captain: tianyipeng
Hefferon Linear Algebra III: Maps, Representation and Change of BasisTextbook
Chapter Three of Jim Hefferon's *Linear Algebra* is about maps between spaces and how matrices represent them. The goal theorem is where the chapter arrives: two matrices represent the same transformation with respect to different bases exactly when they are similar. That is the hinge of the whole book — it converts the search for a canonical form under similarity into the search for the basis in which a map looks simplest, which is the programme of Chapter Five. The milestones are the chapter's landmarks: dimension classifies spaces up to isomorphism, rank plus nullity recovers the dimension of the domain, matrix multiplication is exactly composition, and Gram-Schmidt splits a space into a subspace and its orthogonal complement.
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Captain: tianyipeng
Hefferon Linear Algebra II: Dimension and RankTextbook
Chapter Two of Jim Hefferon's *Linear Algebra* builds the vector space vocabulary — spanning, independence, basis — and turns it into a theory of dimension. The goal theorem is the chapter's most striking result, that the row rank and the column rank of a matrix always agree, which is the bridge between the matrix-of-numbers view of Chapter One and the vector space view of Chapter Two. The milestones are the two pillars it stands on: that any two bases of a space have the same size, so dimension is well defined at all, and that any linearly independent set can be extended to a basis.
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Captain: tianyipeng
Hefferon Linear Algebra I: Gauss's Method and the Solution SetTextbook
Chapter One of Jim Hefferon's *Linear Algebra* develops Gauss's method and asks what row reduction actually preserves. The answer arrives as the Linear Combination Lemma: row operations change the rows of a matrix but never the subspace those rows span, and that invariant is complete. The goal theorem is that completeness — two matrices are row equivalent exactly when they have the same row space — which is what makes reduced echelon form a genuine canonical form. The milestones are the two results the chapter builds on the way: that row operations leave a system's solution set alone, and that a solution set is always one particular solution translated by the solutions of the associated homogeneous system.
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Captain: Community (Bot)
The Jacobian ConjectureOpen Problem
First raised for two variables by Ludwig Kraus in 1884 and stated in full generality by Ott-Heinrich Keller in 1939, the Jacobian conjecture asks something that sounds almost like freshman calculus: if a polynomial map from complex n-space to itself has a Jacobian determinant equal to a nonzero constant, must it be invertible by another polynomial map? That constant-Jacobian condition is precisely the algebraic shadow of the inverse function theorem, yet producing a polynomial — not merely analytic — inverse has resisted every attack for over eighty years. Shreeram Abhyankar championed the problem because it can be stated 'using little beyond a knowledge of calculus,' and Stephen Smale placed it sixteenth on his 1998 list of problems for the new century. Its notoriety is sharpened by a graveyard of published 'proofs' that later collapsed. Deep reductions exist — Bass, Connell, and Wright showed in 1982 that the general case reduces to maps of degree three — and the problem is equivalent, through work of Tsuchimoto, Belov-Kanel, and Kontsevich, to the Dixmier conjecture on the Weyl algebra. A formal statement anchors this famously slippery problem so that progress can be verified rather than merely believed.
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