## Degree of fields query

I'm not sure if my reasoning below is correct or not.

If a=e$\stackrel{\underline{2πi}}{5}$, then Q(a) = {r + sa + ta2 + ua3 +va4 : r,s,t,u,v $\in$ Q} . [Is this correct?]

Then [Q(a):Q] = 5 as {1, a, a2, a3, a4} form a basis for Q(a) as a vector space over Q.

However I am not sure if my reasoning above is correct as I have just seen a proof that [Q(a):Q] = 4 for the same a above.

 Quote by Ad123q I'm not sure if my reasoning below is correct or not. If a=e$\stackrel{\underline{2πi}}{5}$, then Q(a) = {r + sa + ta2 + ua3 +va4 : r,s,t,u,v $\in$ Q} . [Is this correct?] *** No, because $\deg_{\mathbb Q}\mathbb Q(a)=\phi(5)=4$ , so any basis has only 4 elements and not 5, as you wrote. The minimal pol. of $a$ over the rationals is $x^4+x^3+x^2+x+1$ . DonAntonio Then [Q(a):Q] = 5 as {1, a, a2, a3, a4} form a basis for Q(a) as a vector space over Q. However I am not sure if my reasoning above is correct as I have just seen a proof that [Q(a):Q] = 4 for the same a above. Thanks for your help.