Is the Degree of Q(a) Over Q Correctly Understood?

[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?]

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.

 

[DonAntonio]

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.

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