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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 \( \varphi : R[x] \to R[x]/(f(x)) \) 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 \( \varphi \) an isomorphism of groups or rings? It's an isomorphism of rings
2. Complex ring, \( \mathbb{C} \)

There is a \( f(x) \) so that \( \varphi : R[x] \to R[x]/(f(x)) \) makes \( R[x]/(f(x)) \) isomorphic to \( \mathbb{C} \) 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 \( f(x) \in R[x] \) a polynomial. Let \( \varphi: R[x] \to R^{\bigoplus d} \) be a set function defined by by sending \( g(x) \in R[x] \) to the tuple representation of the remainder of \( g(x) \) when divided by \( f(x) \).

Then this function induces an isomorphism of abelian groups:

\[\frac{R[x]}{(f(x))} \cong R^{\bigoplus d} \]