Ideals and polynomials

Forum for the GRE subject test in mathematics.
Hom
Posts: 39
Joined: Sat Oct 01, 2011 3:22 am

Ideals and polynomials

Postby Hom » Sun Oct 16, 2011 3:38 am

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Last edited by Hom on Sun Oct 16, 2011 9:35 am, edited 1 time in total.

owlpride
Posts: 204
Joined: Fri Jan 29, 2010 2:01 am

Re: Ideals and polynomials

Postby owlpride » Sun Oct 16, 2011 3:53 am

(1) and (3):

1+ x^4 = (1+x^2)^2 \text{ (mod 2)}
1 + x^6 = (1+x^3)^2 \text{ (mod 2)}

(2) is not in the ideal because you cannot get an x by multiplying or adding/subtracting powers of x^2 and x^3.

Hom
Posts: 39
Joined: Sat Oct 01, 2011 3:22 am

Re: Ideals and polynomials

Postby Hom » Sun Oct 16, 2011 4:18 am

Thanks owlpride.
Last edited by Hom on Sun Oct 16, 2011 9:38 am, edited 2 times in total.

talkloud
Posts: 17
Joined: Thu Apr 28, 2011 9:44 pm

Re: Ideals and polynomials

Postby talkloud » Sun Oct 16, 2011 8:07 am

(1+x^2)^2\equiv 1 + 2x^2 + x^4 \pmod 2

Since 2x^2 is a multiple of 2, it is congruent to zero mod 2, so you get 1+x^4.

The elements x^2,x^3 work the same way in \mathbb{Z}/2\mathbb{Z}[x] as they do for any other polynomial ring. The construction of R[x] ensures that x is never an element of R, so it will behave the same way regardless of the choice of R.

Hom
Posts: 39
Joined: Sat Oct 01, 2011 3:22 am

Re: Ideals and polynomials

Postby Hom » Sun Oct 16, 2011 9:07 am

That makes prefect sense to me now. Thank you so much for the explanation.

Btw, do you guys know any good problem sets/practices for abstract algebra and general topology? I really think they can get a beginner like me into thinking various problems and getting a better understanding. :D
Last edited by Hom on Sun Oct 16, 2011 9:37 am, edited 1 time in total.

goombayao
Posts: 53
Joined: Sun Oct 16, 2011 9:17 am

Re: Ideals and polynomials

Postby goombayao » Sun Oct 16, 2011 9:19 am

You realize it is against the rules to post this question at all, right?

Hom
Posts: 39
Joined: Sat Oct 01, 2011 3:22 am

Re: Ideals and polynomials

Postby Hom » Sun Oct 16, 2011 9:46 am

goombayao wrote:You realize it is against the rules to post this question at all, right?


Sorry, I was not aware of that but I am now. I've removed the content.

User avatar
miguel
Posts: 11
Joined: Thu Sep 15, 2011 6:51 pm

Re: Ideals and polynomials

Postby miguel » Sun Oct 16, 2011 5:02 pm

^^tattle-tail. WHO CARES.




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