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Can someone tell me, what the generator (in $\mathbb{Z}[X]$) of the ideal $T$ of all polynomials with integer coefficients, such that the first is divisible by $20$ and the second and the third are divisible by $4$ is ?

My guess is, it is $(20,4X,X^3)$ since then I can relatively easy show, that this contains my set $T$ - but I'm stuck showing that this is also contained in $T$.

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    @ThomasAndrews Yes, bad formulation: This set is an ideal.2012-10-23

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Let $f(X),g(X),h(X) \in \mathbb Z[X]$, we have, denoting by $p(X)_i$ the coefficient of $X^i$ of a polynomial $p(X) \in \mathbb Z[X]$ \begin{align*} \bigl(20\cdot f(X) + 4X\cdot g(X) + X^3 \cdot h(X)\bigr)_0 &= 20 \cdot f(X)_0\\ \bigl(20 \cdot f(X) + 4X \cdot g(X) + X^3 \cdot h(X)\bigr)_1 &= 20 \cdot f(X)_1 + 4 \cdot g(X)_0\\ \bigl(20 \cdot f(X) + 4X \cdot g(X) + X^3 \cdot h(X)\bigr)_2 &= 20 \cdot f(X)_2 + 4 \cdot g(X)_1\\ \end{align*} So, by definition of $T$, we have $20f(X) + 4Xg(X) + X^3h(X) \in T$, as $f,g,h$ were arbitrary, $(20, 4X, X^3) \subseteq T$.


On the other side, if $p(X) = \sum_{i=0}^n p_iX^i \in T$ is given, we know $20 \mid p_0$, $4\mid p_1$, $4 \mid p_2$. That is, there are $\alpha, \beta, \gamma \in \mathbb Z$ such that $p_0 = 20\alpha$, $p_1 = 4\beta$, $p_2 = 4\gamma$. But then \begin{align*} p(X) &= \sum_{i=0}^n p_i X^i\\ &= p_0 + p_1 X + p_2 X^2 + X^3 \cdot \sum_{i=0}^{n-3} p_{i+3}X^i \\ &= 20 \cdot \alpha + 4X \cdot (\beta +\gamma X)+ X^3 \cdot \sum_{i=0}^{n-3} p_{i+3}X^i\\ &\in (20, 4X, X^3) \end{align*}

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    But that's the other inclusion ... $(20, 4X, X^3) \supseteq T$, each polynomial in $T$ is generated. Added something about that.2012-10-23
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Hint $\, $ Let $\rm\: S = \{\, 20\,c_0\! + 4\,c_1 x + 4\,c_2 x^2\! + x^3 f(x)\ :\ c_i\in \Bbb Z,\ f\in \Bbb Z[x]\}.\:$ Since $\rm\:20,\,4x,\,x^3\in S\:$ we infer $\rm\:(20,4x,x^3)\subseteq (S).\ $ Conversely $\rm\:20\,c_0\in (20),\,\ 4\,c_1 x + 4\,c_2 x^2\in (4x),\,\ x^3f(x)\in (x^3),\:$ therefore $\rm\:(20,4x,x^3)\supseteq (S),\ $ where $\rm(S)\,$ denotes the ideal generated by the elements of $\rm\, S.$