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Thread: Desperate need of help

  1. #1
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    Desperate need of help

    Let p be a prime integer.

    1. Prove that Z_p is an integral domain.

    2. Prove that Z_p[x] is an integral domain.

    I know that if a ring is a ED it is a PID and if it is a PID it is a UFD and if it is a UFD it is an integral domain. So this can be proven by proving ED, PID or UFD. However, I'm stuck. I need a clear explanation/proof.
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  2. #2
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    Quote Originally Posted by pleasehelpme1 View Post
    I know that if a ring is a ED it is a PID and if it is a PID it is a UFD and if it is a UFD it is an integral domain. So this can be proven by proving ED, PID or UFD. .
    You don't really have to do all of that. To show that a ring is an integral domain, you just need to show that it's commutative, have unity, and have no zero divisors.

    1) Let $\displaystyle a,b \in \mathbb{Z}_p$ and $\displaystyle ab=0$. Then $\displaystyle ab=pk$ for some $\displaystyle k \in \mathbb{Z}$, which means that $\displaystyle p|a$ or $\displaystyle p|b$. Therefore, in $\displaystyle \mathbb{Z}_p, a=0$ or $\displaystyle b=0$. $\displaystyle \Box$



    2) $\displaystyle \mathbb{Z}_p[x]$ is a commutative ring with unity; therefore, we need to show that there are no zero divisors.

    Let $\displaystyle f(x), g(x) \in \mathbb{Z}_p[x]$, where

    $\displaystyle f(x)=a_mx^m+a_{m-1}x^{m-1}+\dots+a_0$
    $\displaystyle g(x)=b_nx^n+b_{n-1}x^{n-1}+\dots+b_0$,

    and $\displaystyle a_m \not=0$ and $\displaystyle b_n \not=0$. Then $\displaystyle f(x)g(x)$ has leading coefficient $\displaystyle a_mb_n$, but since $\displaystyle \mathbb{Z}_p$ is an integral domain, $\displaystyle a_mb_n \not=0$. $\displaystyle \Box$
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  3. #3
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    ....

    can you help me prove that they are both unique factorization domains?
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  4. #4
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    For that one, I think it would be easier to prove that they are both PID's.


    1) $\displaystyle \mathbb{Z}_p $ is a field, and the only ideals in a field is {0} and the field itself. Since {0} $\displaystyle =(0)$ and $\displaystyle \mathbb{Z}_p=(1)$ , $\displaystyle \mathbb{Z}_p$ is a PID. $\displaystyle \Box$

    2) Let's look at the stronger case: if $\displaystyle K$ is a field, then $\displaystyle K[x]$ is a PID.

    Let $\displaystyle I$ be a nontrivial ideal in $\displaystyle K[x]$. Let $\displaystyle g(x)$ be an element in $\displaystyle I$ with minimum degree. We claim that $\displaystyle I=(g(x))$.

    Clearly, $\displaystyle (g(x))\subseteq I$. Let $\displaystyle f(x) \in I$. We can write $\displaystyle f(x)$ as $\displaystyle f(x)=g(x)q(x)+r(x)$, where $\displaystyle r(x)=0$ or deg $\displaystyle r(x)$< deg $\displaystyle g(x)$.
    Solving for $\displaystyle r(x)$ gives us $\displaystyle r(x)=f(x)-g(x)q(x) \in I \Longrightarrow r(x)=0$ (minimality of $\displaystyle g(x)$). Therefore, $\displaystyle f(x)=g(x)q(x) \in (g(x)) \Longrightarrow I \subseteq (g(x))$. $\displaystyle \Box$
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  5. #5
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    ??

    how do you know the only ideal are {0} and the field itself? and how does this mean that every ideal can be written as <a> for some a in the field?
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  6. #6
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    Quote Originally Posted by pleasehelpme1 View Post
    how do you know the only ideal are {0} and the field itself?
    Well, you use the fact that if an ideal of a ring contains unity, then the ideal is equal to the ring. Every nonzero element in a field is a unit, so it pretty much follows from there.
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  7. #7
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    ??

    you are completely losing me here. can you explain this in a simpler way? isn't a field a PID only if every ideal is generated by only one ideal in the field? if so how is {0} and the field itself generated by the same ideal?
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  8. #8
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    No, {0} and the field isn't generated by the same element.

    Ok, let's say we have a field $\displaystyle F$ and a nontrivial ideal $\displaystyle I$ of $\displaystyle F$. Let $\displaystyle a$ be an element in $\displaystyle I$. Since $\displaystyle a$ is also in $\displaystyle F$, there exists an element $\displaystyle b$ in $\displaystyle F$ such that $\displaystyle ab=1$ (every nonzero element in a field is a unit). Then, by the definition of an ideal, $\displaystyle 1=ab \in I$. Since $\displaystyle I$ contains unity, $\displaystyle I=F$. Therefore, the only ideals in the field $\displaystyle F$ are {0} and $\displaystyle F$.

    In the case of $\displaystyle \mathbb{Z}_p$, 1 is a generator for $\displaystyle \mathbb{Z}_p$, so $\displaystyle \mathbb{Z}_p=(1)$, and clearly {0} $\displaystyle =(0)$.
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  9. #9
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    ....

    so the its that each ideal is generated by a single element? not that its the same elements that generates all ideals? also how do we know that all nonzero elements in a field are units?
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  10. #10
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    ???

    forget about my last question, but does this mean that the only ideals of all fields are {0} and the field itself?
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