[Verse 1]
Ten courses carved a pathway through the maze
Linear algebra builds the foundational base
Vectors dance in spaces, transformations flow
Maps between dimensions, watch the patterns grow
Decompositions break apart what seems complex
Concrete structures waiting to be explored next
[Chorus]
Spaces, maps, and symmetry divine
Groups and rings in perfect design
Fields to groups through Galois' bridge
Modules climbing every ridge
Categories weaving all the threads
This is where abstraction spreads
[Verse 2]
Symmetry gets captured in the language of groups
Abstract essence spinning through algebraic loops
Rings and fields extend arithmetic's domain
Two operations dancing, breaking every chain
Galois built a bridge between these separate worlds
Impossibility theorems like flags unfurled
[Chorus]
Spaces, maps, and symmetry divine
Groups and rings in perfect design
Fields to groups through Galois' bridge
Modules climbing every ridge
Categories weaving all the threads
This is where abstraction spreads
[Verse 3]
Modules generalize what vectors used to be
Over rings instead of fields, exact sequences free
Commutative algebra paints geometry's face
Ideals correspond to varieties in space
Homological measures where exactness breaks
Shows us all the places where precision shakes
[Bridge]
Representations make the groups come alive
Acting on the spaces where geometries thrive
Lie algebras capture continuous flow
Periodic tables of symmetry grow
Category theory sees the bigger game
Mathematics studying its own domain
[Verse 4]
From concrete linear structure to abstract height
Every course illuminates another kind of sight
Structure itself becomes the final goal
Patterns within patterns, mathematics whole
Ten stepping stones across the algebraic stream
Building up the language of the abstract dream
[Outro]
What this path teaches runs deeper than technique
It's the grammar of the mathematical mystique
From vectors up to categories above
This is algebra's architectural love