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

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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

(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

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

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

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

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.
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

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

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.

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

Re: Ideals and polynomials

^^tattle-tail. WHO CARES.