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Can anyone help me with this problem. I try many ways to solve this but still can't find the solution.Thanks

Let $K$ be a subfield of field $F$, and suppose $f$, $g$ are polynomials in $K[x]$. Let $M_k$ be the ideal generated by $f$ and $g$ in $K[x]$ and $M_f$ be the ideal they generate in $F[x]$.Show that $M_k$ and $M_f$ have the same monic generator

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    Can you clarify more? What does PID mean?2012-12-11

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The ideal $M_f$ is generated by some greatest common divisor $\gcd(f,g)$ of $f$ and $g$ in $F[x]$ . But if you look at the algorithm for computing the $\gcd(f,g)$ you will find that only the coefficients of $f$ and $g$ are used in the calculations. In particular, if $f$ and $g$ lie in $K[x]$ then the same holds for $\gcd(f,g)$.