Opened 6 years ago

Last modified 5 years ago

## #21996 closed enhancement

# Factorization in iterated extensions of finite fields — at Initial Version

Reported by: | Julian Rüth | Owned by: | |
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Priority: | major | Milestone: | sage-7.5 |

Component: | finite rings | Keywords: | factorization, finite field, sd86.5, sd87 |

Cc: | Jean-Pierre Flori | Merged in: | |

Authors: | Julian Rüth | Reviewers: | |

Report Upstream: | N/A | Work issues: | |

Branch: | Commit: | ||

Dependencies: | Stopgaps: |

### Description

At the moment there is no factorization implemented in iterated extensions of finite fields:

sage: K = GF(2) sage: R.<x> = K[] sage: L.<x> = K.extension(x^2 + x + 1) sage: R.<y> = L[] sage: M.<y> = L.extension(y^2 + y + x) sage: R.<T> = M[] sage: (T^2 + T + x).factor()

The reason is that `M`

in the above example is just a quotient of a polynomial ring over `L`

.
Here, we implement isomorphisms (for this special case) to a simple extensions of a prime field over which factorization is implemented. We also implement univariate factorization for polynomial quotient rings if an isomorphism to a field which supports such factorization is known.

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