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Synku, z Bogiem, powodzenia na egzaminach nerwowo żegnała rano Bogumiła swojego jedynego syna Czesława, gdy wychodził do akademika.

Miał jechać na egzaminy wstępne do sąsiedniego miasta. Pokrzepiła go śniadaniem, dodała otuchy i wyprawiła, by zdobywał nowe szczyty.

Dzięki, mamo, nie martw się, będzie dobrze. Na budżet pewnie się nie dostanę za Czesławem zatrzasnęły się drzwi, a mąż już dawno wyjechał do pracy.

Bogumiła z Wojciechem przeżyli razem dwadzieścia dwa lata. Wychowali syna Czesława na por# Issue 15753: Wrong result in polynomial division in GF(2)[x]

archive/issues_015516.json:
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\n\nI was trying to compute the minimal polynomial of a certain matrix mod 2 (i.e. in GF(2)), and got a wrong result. After some investigation it seems that the problem is in polynomial division in GF(2)[x]. In particular, the following code (which is the step that seems to fail in the minimal polynomial computation) gives a wrong result:\n\n„`\nsage: ring. = GF(2)[]\nsage: f = x^14 + x^13 + x^12 + x^10 + x^9 + x^7 + x^6 + x^4 + x^2 + x\nsage: g = x^8 + x^6 + x^5 + x^4 + x^2 + x + 1\nsage: q, r = f.quo_rem(g)\nsage: f == q*g + r\nFalse\n„`\n\nBut if I use the generic polynomial division code, the result is correct:\n\n„`\nsage: q1, r1 = f._quo_rem_naive(g)\nsage: f == q1*g + r1\nTrue\n„`\n\nComponent: **algebra**\n\nKeywords: **polynomial division, GF(2)**\n\nAuthor: **Peter Bruin**\n\nBranch/Commit: **[`41e5a23`](https://github.com/sagemath/sagetrac-mirror/commit/41e5a23d4d1b6a3f2f70761c0b8e8b0c9db960a5)**\n\nReviewer: **Mariah Lenox**\n\n_Issue created by migration from https://trac.sagemath.org/ticket/15753_\n\n”,
„closed_at”: „2014-03-31T21:13:11Z”,
„created_at”: „2014-01-28T13:57:03Z”,
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„repository”: „https://github.com/sagemath/sage”,
„title”: „Wrong result in polynomial division in GF(2)[x]”,
„type”: „issue”,
„updated_at”: „2014-03-31T21:13:11Z”,
„url”: „https://github.com/sagemath/sage/issues/15753”,
„user”: „https://github.com/pjbruin”
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„`

I was trying to compute the minimal polynomial of a certain matrix mod 2 (i.e. in GF(2)), and got a wrong result. After some investigation it seems that the problem is in polynomial division in GF(2)[x]. In particular, the following code (which is the step that seems to fail in the minimal polynomial computation) gives a wrong result:

„`
sage: ring. = GF(2)[]
sage: f = x^14 + x^13 + x^12 + x^10 + x^9 + x^7 + x^6 + x^4 + x^2 + x
sage: g = x^8 + x^6 + x^5 + x^4 + x^2 + x + 1
sage: q, r = f.quo_rem(g)
sage: f == q*g + r
False
„`

But if I use the generic polynomial division code, the result is correct:

„`
sage: q1, r1 = f._quo_rem_naive(g)
sage: f == q1*g + r1
True
„`

Component: **algebra**

Keywords: **polynomial division, GF(2)**

Author: **Peter Bruin**

Branch/Commit: **[`41e5a23`](https://github.com/sagemath/sagetrac-mirror/commit/41e5a23d4d1b6a3f2f70761c0b8e8b0c9db960a5)**

Reviewer: **Mariah Lenox**

_Issue created by migration from https://trac.sagemath.org/ticket/15753_

—

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