Factoring a polynomial over a finite Field

asked 2014-12-04 09:29:43 +0200

fagui
187 ●9 ●16 ●22

updated 2016-06-06 20:02:29 +0200

slelievre
17654 ●22 ●160 ●348 http://carva.org/samue...

Hello

I'm following a 101 algebra course, and for example, I would like to factor a polynomial on a finite field like F_9 (F_9 == ZZ/9ZZ is a field because 9 is a power of a prime number, 3)

R = PolynomialRing(GF(9),'x')
x = R.gen()
f = x^4+x^2-1
f in R
f.factor()

i get an error message

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_68.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("UiA9IFBvbHlub21pYWxSaW5nKEdGKDkpLCd4JykKeCA9IFIuZ2VuKCkKZiA9IHheNCt4XjItMQpmIGluIFI="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>

  File "/tmp/tmpYgKq_W/___code___.py", line 3, in <module>
    R = PolynomialRing(GF(_sage_const_9 ),'x')
  File "factory.pyx", line 364, in sage.structure.factory.UniqueFactory.__call__ (build/cythonized/sage/structure/factory.c:1244)
  File "/home/sage/sage-6.3/local/lib/python2.7/site-packages/sage/rings/finite_rings/constructor.py", line 414, in create_key_and_extra_args
    raise ValueError("parameter 'conway' is required if no name given")
ValueError: parameter 'conway' is required if no name given

i'm running sage 6.3 notebook on windows through Oracle VM winbox. I'm a totally new user, and i've looked at the tutorial and the forum but couldn't find any example or reason why this would not work. thank you for your help.

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To display code in your question, select it and click the "code" button (the one with 010 101).

slelievre ( 2014-12-06 17:30:08 +0200 )edit

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answered 2014-12-04 10:32:29 +0200

slelievre
17654 ●22 ●160 ●348 http://carva.org/samue...

updated 2016-06-06 20:04:40 +0200

The error in your math is that the group and ring structures on F9 are not the same as those on Z/9Z. As you know, Z/nZ is a field only when n is prime.

The error in your code comes from calling GF(9). You get the same error by just this:

sage: GF(9)
...
ValueError: parameter 'conway' is required if no name given

Instead, you need to define the generator at the time you instantiate GF(9). Do this:

sage: F.<a> = GF(9)

Then F is defined, with a as a generator (over Z/3Z). You can check its minimal polynomial:

sage: p = a.minpoly()
sage: p 
x^2 + 2*x + 2
sage: p.parent()
Univariate Polynomial Ring in x over Finite Field of size 3

The rest of your code should work from there.

sage: R.<x> = PolynomialRing(F)
sage: f = x^4 + x^2 - 1
sage: f in R
True
sage: f.factor()
(x^2 + a) * (x^2 + 2*a + 1)

Edit: In Sage 7.2 you can call GF(9) without specifying a generator name or a polynomial. Your original code works in Sage 7.2.

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thank you very much. yes i realized that shortly after posting !

fagui ( 2014-12-06 09:23:16 +0200 )edit

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