## #11057 closed defect (invalid)

# GCD in Polynomial Rings over Extension Fields

Reported by: | Chris Swierczewski | Owned by: | Alex Ghitza |
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Priority: | major | Milestone: | sage-duplicate/invalid/wontfix |

Component: | algebra | Keywords: | gcd, polynomialring, extension fields |

Cc: | Merged in: | ||

Authors: | cswiercz | Reviewers: | cswiercz |

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

Branch: | Commit: | ||

Dependencies: | Stopgaps: |

### Description

During an `ask.sagemath.org`

response (see symbolic polynomial euclidean algorithm) I came across a bug in computing the GCD of two elements of a polynomial ring over an extension field. For example:

sage: R.<x> = PolynomialRing(QQ,'x') sage: p = x^2-2 sage: q = x^2-3 sage: K.<a,b> = QQ.extension([p,q]) sage: S.<x> = PolynomialRing(K,'x') sage: f = a*x^2 * (x-1); g = a*x^2 * (x-b) sage: f.gcd(g) x^2

However, the result should be `a*x^2`

.

### Change History (2)

### comment:1 Changed 12 years ago by

Resolution: | → invalid |
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Reviewers: | → cswiercz |

Status: | new → closed |

### comment:2 Changed 12 years ago by

Milestone: | sage-4.7 → sage-duplicate/invalid/wontfix |
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**Note:**See TracTickets for help on using tickets.

Mathematical mistake.

`a`

`is a unit in $QQ(a,b)$ so it's not part of the GCD. This ticket is not an issue / defect.`