[Verse 1] We start with vectors in a space so wide Linear combinations where solutions hide Matrices multiply in ordered rows Eigenvalues show us where the system goes From basic operations to inner products true The spectral theorem breaks it down for you [Chorus] Algebra paths ascending high Groups and rings and fields that fly From linear maps to categories Mathematical symphonies Structure builds on structure strong This is our algebra song [Verse 2] Groups have elements with one operation Closure, inverse, and association Subgroups living in a larger frame Homomorphisms keep the structure same Lagrange tells us about the order's way Quotient groups partition and display [Chorus] Algebra paths ascending high Groups and rings and fields that fly From linear maps to categories Mathematical symphonies Structure builds on structure strong This is our algebra song [Verse 3] Rings have two operations now in play Addition, multiplication show the way Ideals are subsets with absorption power Fields where every nonzero has its hour Polynomials extend the field we know Extensions let the algebraic garden grow [Bridge] Galois theory unlocks the ancient quest Which equations can be solved and which can't pass the test Automorphisms dance between the fields Fundamental theorem shows what solvability yields [Verse 4] Modules generalize the vector theme Over rings they build the abstract dream Exact sequences keep the maps aligned Tensor products weave what's intertwined Commutative rings with Noetherian chains Localization focuses where structure remains [Verse 5] Homology counts the holes and gaps between Chain complexes make the patterns seen Representations show how groups can act Characters reveal the abstract fact Lie algebras with their bracket laws Root systems following structure's cause [Chorus] Algebra paths ascending high Groups and rings and fields that fly From linear maps to categories Mathematical symphonies Structure builds on structure strong This is our algebra song [Outro] Categories connect with functors bright Natural transformations in the light Adjunctions show the deeper harmony Yoneda lemma sets the structures free From vectors up to categories We've climbed the algebra trees
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