[Verse 1]
When Galois meets your field extension
Count the order, feel the tension
Group size equals degree precisely
Mathematics dancing so concisely
Subgroups map to fixed field treasures
Hidden symmetries beyond all measures
[Chorus]
Galois bridges, building connections
Between the groups and field reflections
Solvable radicals, constructible dreams
Nothing's quite the way it seems
Order equals degree, that's the key
In this algebraic symphony
[Verse 2]
Take a subgroup, find what's stable
Elements fixed upon the table
Intermediate fields climb the ladder
Each one makes the Galois fatter
Normal subgroups hold the secret
Galois quotients, so aesthetic
[Chorus]
Galois bridges, building connections
Between the groups and field reflections
Solvable radicals, constructible dreams
Nothing's quite the way it seems
Order equals degree, that's the key
In this algebraic symphony
[Bridge]
Polynomials whisper their fate
Solvable groups decide their state
Symmetric groups know the limit
Four's the max, don't try to spin it
Constructible points need powers of two
Compass and straightedge will see you through
[Verse 3]
Normal subgroup means quotient's clean
Galois extension, pure and lean
Radical towers built with care
Solvable structure floating in air
Degree must be two to some power
For construction in that crucial hour
[Chorus]
Galois bridges, building connections
Between the groups and field reflections
Solvable radicals, constructible dreams
Nothing's quite the way it seems
Order equals degree, that's the key
In this algebraic symphony
[Outro]
From quintic's impossible quest
To ruler's geometric test
Galois theory shows the way
Through abstract algebraic play