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Math and science::Algebra::Aluffi

Two new rings from polynomial ring quotients

Using the below theorem:

1. Polynomial evaluation

There is a f(x) so that φ:R[x]→R[x]/(f(x)) sends any polynomial g(x) to g(a).

  1. What is f(x)? f(x)=x−a
  2. What is R[x]/(f(x))? It is R
  3. Is φ an isomorphism of groups or rings? It's an isomorphism of rings
2. Complex ring, C

There is a f(x) so that φ:R[x]→R[x]/(f(x)) makes R[x]/(f(x)) isomorphic to C as a ring.

  1. [What is f(x)?]
  2. [How is the multiplication operation transferred to R[x]/(f(x))?]

Let R be a commutative ring, R[x] a polynomial ring, and f(x)∈R[x] a polynomial. Let φ:R[x]→R⨁d be a set function defined by by sending g(x)∈R[x] to the tuple representation of the remainder of g(x) when divided by f(x).

Then this function induces an isomorphism of abelian groups:

R[x](f(x))≅R⨁d