I have these two doubts, which are scattered only couple of pages away in Artin's book, so I am asking them in one post only:
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Artin in his book algebra first shows that $(x^3-2)$ is kernel of substitution homomorphism $\Phi:\Bbb Q[x]\to\Bbb C$ given by $x\to\sqrt[3]2$ is $(x^3-2)$. Then claims that restriction $\Phi'$ of $\Phi$ to $\Bbb Z[x]$ has kernel that is also generated by $(x^3-2)$, and as a proof of that quotes this theorem: Let $f$ be a monic integer polynomial. If $f$ divides $g$ in $\Bbb Q[x]$, then $f$ divides $g$ in $\Bbb Z[x]$.
I can't see why restriction $\Phi':\Bbb Z[x]\to\Bbb C$ has ideal $(x^3-2)$ from above argument.
Also, does above argument has to do something with "Let $f(x) \in \Bbb Z[x]$. If $f(x)$ is reducible over $\Bbb Q$, then it is reducible over $\Bbb Z$. " ? -
They give this example: "We identify the ideals of quotient ring $R'=\Bbb C[t]/(t^2-1)$ using the canonical homomorphism $\pi:\Bbb C[t]\to R'$. The kernel of $\pi$ is the principal ideal $(t^2-1)$. Let $I$ be an ideal of $\Bbb C[t]$ that contains $t^2-1$. Then $I$ is principal, generated by monic polynomial $f$, and the fact that $t^2-1$ is in $I$ means that $f$ divides $t^2-1$. The monic divisors of $t^2-1$ are $1,t-1,t+1,t^2-1$. Therefore the ring $R'$ contains exactly four ideals. They are the principal ideals generated by the residues of the divisors of $t^2-1$."
Please explain what is meant by 'residues of the divisors'.
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