Blog post unifies polynomial arithmetic (division, CRT) with linear algebra over vector spaces via quotient spaces and dimension-counting.
Key Takeaways
Treats polynomials divisible by p as a subspace Wp, then shows V/Wp has dimension equal to deg(p), recovering polynomial division cleanly.
Proves V = Wp + Rp (direct sum) purely from a dimension-matching lemma, no polynomial-specific machinery needed.
Derives the Chinese Remainder Theorem for polynomials from a general quotient decomposition: V/intersect(Wi) isomorphic to product of V/Wi when dimensions sum correctly.
The canonical map g is injective by construction; invertibility follows from dimension equality alone, making CRT constructive via Bezout’s lemma.
References Wisbauer’s module theory and Fuhrmann’s polynomial approach to linear algebra; exercises throughout target readers comfortable with abstract algebra basics.
