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let $a$ a complex number , and $f$ be an irreducible polynomial with integer coefficients such that : $ f(a)=0$

  • 1) Show that the set : $\{ g(a) \mid g \in \mathbf{Z}[X]\}$ is a ring isomorphic to $\mathbf{Z}^{n}$ respect to their group structure where $n=\deg f$

  • 2) show that every non null Ideal of the previous ring is isomorphic to $\mathbf{Z}^{n}$ respect to their group sturcture

Help me to solve these questions with some hints please :)

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    Is $f$ in $\mathbf Q[X]$? What does $\mathbf Q(a)/\mathbf Q \in \mathbf Z[X]$ mean?2011-07-23
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    @mathfan: "Show that $Q(a)/Q\in Z[X]$ is a ring isomorphic to..." What?! First, what is $Q$ and what is $Q(a)$? Second, if $Q(a)/Q$ is an element of $Z[X]$, then it's a polynomial, not a ring.2011-07-23
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    @ arturo Magidin : sorry it is a latex problem : the set { Q(a)/Q \in Z[X]} is the set of images of a by all polynomials in Z[X]2011-07-23
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    @mathfan: No, it's a concept problem: if something is an *element* of a polynomial ring, then chances are that it is a polynomial, **not** a ring itself. What set? What is $Q$? (To get `{` and `}` to show up, you need to use `\{` and `\}`, but even with them, $\{Q(a)/Q \in Z[x]\}$ is still confused and confusing).2011-07-23
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    @mathfan Ah. This is commonly denoted $\mathbf Z[a]$. In this case, you probably want $f$ to have integer coefficients.2011-07-23
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    Ok it must be a misuderstanding , because I'm french and I use french notations , I meant by this set : Z[a] which is the ring generated by : a^0=1,a,a²,a^3... I hope you understand me and I apologize again2011-07-23
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    You also want $f$ to be monic, I think (What is the rank of $\mathbf Z[\frac 12]$?).2011-07-23
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    This question is in my opinion hopelessly garbled. The OP needs to check his/her sources and ask a much clearer question. As Dylan has pointed out, even with all the help given by others in clarifying the question, it seems to be false as stated: e.g. it is false for $f(x) = 2x-1$. I have voted to close.2011-07-23
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    @mathfan: Being French has nothing to do with your plight in this case. First you had problems with TeXifying. That's ok, if you have never done that. But you are still leaving out an awful lot of assumptions. We could try an guess that you 1) want $f$ to be monic (as Dylan suggested), 2) the ring operations that you left out are those of the complex numbers, 3) you are *only* referring to the additive structure in both your questions. Please edit or this may end up in the waste basket.2011-07-23

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