Basic Arithmetic

[:days4/projects/: Other SAGE Days 4 Project]

Test cases Read these as "an element of S and an element of R"

  1. \mathbf{Z}[x]/\mathbf{Z} \in \mathbf{Q}[x] (not Frac(\mathbf{Z}[x]))

  2. \mathbf{Q} + \mathbf{Z}[x] \in \mathbf{Q}[x] and \mathbf{Z}/5\mathbf{Z} + \mathbf{Z}[x] \in \mathbf{Z}/5\mathbf{Z}[x]

  3. \mathbf{Q} * \mathbf{Z}[x] \in \mathbf{Q}[x]

  4. \mathbf{Q} * \mathbf{Z}/5\mathbf{Z} error due to no morphism from all of \mathbf{Q} into \mathbf{Z}/5\mathbf{Z}.

  5. \mathbf{Z}[x] + \mathbf{Z}[y] error due to unknown relation between x and y and ambiguous order

  6. \mathbf{Q}[\zeta_m] + \mathbf{Q}[\zeta_n] \in \mathbf{Q}[\zeta_{lcm(m,n)}] as cyclotomic fields are created with an embedding into \bar{\mathbf{Q}}

  7. \mathbf{F}_{p^n} + \mathbf{F}_{p^m} works using Conway polynomials

  8. \mathbf{Z}[x] + \mathbf{Q}[y] same as 5 (minus the symmetry concerns)

  9. \mathbf{Z}/16\mathbf{Z} + \mathbf{Z}/12\mathbf{Z} \in \mathbf{Z}/4\mathbf{Z}

Proposed model

One can view most desired natural coersions as functorial operations from some simpler base object. E.g.

Given two objects S and R, first see if there is a natural morphism S \rightarrow R or R \rightarrow S. If there is one, use it. If there are two, by definition they are cannonically isomorphic and so we can use either.

Considering functors above, decompose each object R as a tower of simpler rings R^0 \rightarrow \cdots \rightarrow R'' \rightarrow R' \rightarrow R. Now try to complete the pushout diagram R \rightarrow Y \leftarrow R for some Y as low as possible in the tower of R and S. If the pushout exists, do arithmatic in this ring.

Otherwise we search for a

Other