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Hiall,

I am asked to show that if $L=K(\beta)$, prove that the minimal polynomial of $\beta$ is the characteristic polynomial of $T(\beta)$, with leading coefficient 1 (monic).


This is part (iii) of a question, part (i) and (ii) I have done, and I list it below in case it is needed for part (iii)

(i) Assume $[L:K]=n$. For any $\beta\in L$, show that the map $T_\beta(k)=\beta k$ is a linear transformation (scalars in K) on the vector space L. Hence, show that $L$ is isomorphic to a subring of $M_n(K)$.

(ii) Show that $Tr(\beta)=Trace(T(\beta))$ and $N(\beta)=det(T(\beta))$ are additive and multiplicative (respectively) group homomorphisms.

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Can you see that if $p$ is any polynomial with coefficients in $K$ then $p(T_{\beta})(k)=p(\beta)k$? –  Gerry Myerson Sep 5 '12 at 13:16
    
Does that follow from the definition $T_\beta(k):=\beta k$? –  yoyostein Sep 5 '12 at 13:19
    
Yes. E.g., $T^2(k)=T(T(k))=T(\beta k)=\beta\beta k=\beta^2k$. –  Gerry Myerson Sep 5 '12 at 13:24
    
Thanks for the comment! –  yoyostein Sep 6 '12 at 0:59
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