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I need help proving this theorem:

Given the field extension: $\mathbf{K} \subseteq \mathbf{L}$, for $\alpha \in \mathbf{L}$ and $g(x) \in \mathbf{K}[x]$, $\alpha$'s minimal polynomial over $K$, and $f(x) \in \mathbf{L}[x]$, $\alpha$'s minimal polynomial over $L$, then the degree of $g$ is bigger than the degree of $f$ and $f(x)$ divides $g(x)$.

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    $\begingroup$ As you assume $\alpha\in \mathbf L$, isn't automatically $f(x)=x-\alpha$? $\endgroup$ Jul 14, 2013 at 14:55
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    $\begingroup$ Sorry, I meant $\mathbf{K} \subseteq \mathbf{L} \subseteq \mathbf{M}$ and $\alpha \in \mathbf{M}$. $\endgroup$
    – Daniella
    Jul 14, 2013 at 15:04
  • $\begingroup$ Of course we cannot conclude from the stated facts that degree of $g$ is strictly "bigger" than degree of $f$. Indeed your assumptions allow $\mathbf{K} = \mathbf{L}$, so we cannot rule out the possibility $g=f$. $\endgroup$
    – hardmath
    Jul 14, 2013 at 15:36

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Because $\mathbf{K}\subseteq\mathbf{L}$, you also have $\mathbf{K}[x]\subseteq\mathbf{L}[x]$, so that $g\in \mathbf{L}[x]$ and $g(\alpha)=0$, and therefore (because $f$ is the minimal polynomial of $\alpha$ over $\mathbf{L}$) we must have $f\mid g$, and hence also $\deg(f)\leq\deg(g)$.

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  • $\begingroup$ You go straight from "what we assume" to "what we want to be true". "Because $f$ is the minimal polynomial of $\alpha$ over $L$" can have many implications. Not all of them will get us where we want to be, so be specific: which properties of the minimal polynomial are you using? $\endgroup$ Apr 5, 2020 at 22:10
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    $\begingroup$ The property that "if $g\in \mathbf{L}[x]$ has $g(\alpha)=0$, then the minimal polynomial of $\alpha$ over $\mathbf{L}$ must divide $g$" - in other words, the property that the minimal polynomial is a generator of the ideal $\{g\in \mathbf{L}[x]:g(\alpha)=0\}$ in the ring $\mathbf{L}[x]$. $\endgroup$ Apr 6, 2020 at 16:49
  • $\begingroup$ math.stackexchange.com/questions/3780912/… $\endgroup$ Aug 5, 2020 at 14:35

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