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Math Help - Factor ring

  1. #1
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    Factor ring

    Let I = <x^2 + 5x +6> = {(x^2 + 5x+6) *f| f in Q[x]} Does the factor ring Q[x]/I have any zero divisors?
    Clearly, x^2 +5x+6 is not irreducible as it equals (x+2)(x+3), thus Q[x]/I is not a field, however I am having difficulty showing whether or not it is an integral domain.
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  2. #2
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    Quote Originally Posted by Coda202 View Post

    Let I = <x^2 + 5x +6> = {(x^2 + 5x+6) *f| f in Q[x]} Does the factor ring Q[x]/I have any zero divisors?
    Clearly, x^2 +5x+6 is not irreducible as it equals (x+2)(x+3), thus Q[x]/I is not a field, however I am having difficulty showing whether or not it is an integral domain.
    (x+2 +I)(x+3 + I)=0 and x+2 +I \neq 0, \ x+3 + I \neq 0. so x+2+I and x+3 +I are zero divisors.
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  3. #3
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    Quote Originally Posted by Coda202 View Post
    Let I = <x^2 + 5x +6> = {(x^2 + 5x+6) *f| f in Q[x]} Does the factor ring Q[x]/I have any zero divisors?
    Clearly, x^2 +5x+6 is not irreducible as it equals (x+2)(x+3), thus Q[x]/I is not a field, however I am having difficulty showing whether or not it is an integral domain.
    If F_1,F_2 are fields then F_1\times F_2 has zero divisors. In fact, all zero divisors are given by: (a,0),(0,b) where a\in F_1^{\times}, b\in F_2^{\times}.

    Notice that x^2 + 5x+6 = (x+2)(x+3). Thus, by Chinese remainder theorem:
    \mathbb{Q}[x]/(x^2+5x+6) \simeq \mathbb{Q}[x]/(x+2) \times \mathbb{Q}[x]/(x+3)\simeq \mathbb{Q}\times \mathbb{Q}
    As you can see the RHS has many many zero divisors.
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