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Thread: primes and polynomials

  1. #1
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    primes and polynomials

    Let p be a fixed prime and let J be the set of polynomials in Z[x] whose constant terms are divisible by p. Is J a maximal ideal in Z[x]? Prove or disprove.

    I think it is, but not sure how to prove it.
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  2. #2
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    Quote Originally Posted by zhupolongjoe View Post
    Let p be a fixed prime and let J be the set of polynomials in Z[x] whose constant terms are divisible by p. Is J a maximal ideal in Z[x]? Prove or disprove.

    I think it is, but not sure how to prove it.

    No, it isn't: for example, $\displaystyle <2>\subset <2,x>$ . The last one is a maximal ideal.

    Tonio

    Ps. You may find the following interesting:
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  3. #3
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    i think tonio misread the question. $\displaystyle J$ is a maximal ideal. the reason is that suppose $\displaystyle J \subset I \subseteq \mathbb{Z}[x],$ for some ideal $\displaystyle I.$ choose $\displaystyle f = a_0 + a_1x + \cdots + a_nx^n \in I \setminus J.$

    since the constant term of $\displaystyle f-a_0$ is zero, we have $\displaystyle f - a_0 \in J \subset I$ and so $\displaystyle a_0=f - (f-a_0) \in I.$ also $\displaystyle \gcd(a_0,p)=1$ because $\displaystyle f \notin J.$ so $\displaystyle ra_0 + sp = 1,$ for

    some inetgers $\displaystyle r,s \in \mathbb{Z}.$ it follows that $\displaystyle 1=ra_0 + sp \in I,$ i.e. $\displaystyle I=\mathbb{Z}[x].$
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    Quote Originally Posted by NonCommAlg View Post
    i think tonio misread the question. $\displaystyle J$ is a maximal ideal. the reason is that suppose $\displaystyle J \subset I \subseteq \mathbb{Z}[x],$ for some ideal $\displaystyle I.$ choose $\displaystyle f = a_0 + a_1x + \cdots + a_nx^n \in I \setminus J.$

    since the constant term of $\displaystyle f-a_0$ is zero, we have $\displaystyle f - a_0 \in J \subset I$ and so $\displaystyle a_0=f - (f-a_0) \in I.$ also $\displaystyle \gcd(a_0,p)=1$ because $\displaystyle f \notin J.$ so $\displaystyle ra_0 + sp = 1,$ for

    some inetgers $\displaystyle r,s \in \mathbb{Z}.$ it follows that $\displaystyle 1=ra_0 + sp \in I,$ i.e. $\displaystyle I=\mathbb{Z}[x].$

    I think this time I didn't misread the question (hurray! 't was about freaking time), so if I did a mistake it was a "honest" one...and I did: the ideal $\displaystyle J$ is NOT the same as the ideal $\displaystyle <p>$ for some prime $\displaystyle p$ ; the latter is the ideal of all pol's all whose coef's are divisible by the prime , whereas the former (the one the OP asked about) is $\displaystyle <p,x>$, which indeed is a maximal ideal. I confused these two.
    Anyway, reading the PDF file I attached to muy first post the OP, hopefully, could have realized the above and overcome my misdirections.

    Tonio
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