I am trying to do the following problem that appears in Dummit and Foote book. But I have no idea how to start the problem. Can anyone please give me a hint? Thank you.
Section 14.5, Problem #1(Page#603): Determine the minimal polynomials satisfied by the primitive generators given in the text for the subfields of $\mathbb{Q(\zeta_{13})}$.
I’ll show you a trick that I use, really just a variation of what’s obvious, but organized so that you can trap careless errors. I promise you that I did this all by hand, no machine aid.
I’ll attack the “simplest” one, finding the minimal polynomial (it has to be quadratic) for $\xi=\zeta+\zeta^3+\zeta^4+\zeta^9+\zeta^{10}+\zeta^{12}$. I’m going to be consistent in rewriting this and all quantities so as to use the fact that $\zeta^{13}=1$. In particular, we have $$ 0=\zeta^6+\zeta^5+\zeta^4+\zeta^3+\zeta^2+\zeta+1+\zeta^{-1}+\zeta^{-2}+\zeta^{-3}+\zeta^{-4}+\zeta^{-5}+\zeta^{-6}\,. $$ And then, calculating, $$ \begin{matrix} \xi=&\zeta^4&+\zeta^3&+\zeta&+\zeta^{-1}&+\zeta^{-3}&+\zeta^{-4}\\ \xi^2=&\zeta^8&+2\zeta^7&+2\zeta^5&+2\zeta^3&+2\zeta&+2\\ &&+\zeta^6&+2\zeta^4&+2\zeta^2&+2&+2\zeta^{-2}\\ &&&+\zeta^2&+2&+2\zeta^{-2}&+2\zeta^{-3}\\ &&&&+\zeta^{-2}&+2\zeta^{-4}&+2\zeta^{-5}\\ &&&&&+\zeta^{-6}&+2\zeta^{-7}\\ &&&&&&+\zeta^{-8}\,, \end{matrix} $$ in which you collect like terms, using $\zeta^8=\zeta^{-5}$ etc., to get $$ \begin{align} \xi^2&=3\zeta^6+3\zeta^5+2\zeta^4+2\zeta^3+3\zeta^2+2\zeta+6+2\zeta^{-1}+3\zeta^{-2}+2\zeta^{-3}+2\zeta^{-4}+3\zeta^{-5}+3\zeta^{-6}\\ &=-\zeta^4-\zeta^3-\zeta+3-\zeta^{-1}-\zeta^{-3}-\zeta^{-4}\\ &=-\xi+3\,, \end{align} $$ in which I subtracted $3$ times zero to go from the first line to the second. And so you see that $\xi^2+\xi-3=0$.