Thread: Cyclotomic Fields & Bases....

Hi everyone! :-)

Suppose Zeta is any primitive 5th root of unity.

Then we know that Q(Zeta) has a basis {1,Zeta,Zeta^2,Zeta^3} because Zeta is algebraic over Q.

But how can I show that {Zeta,Zeta^-1,Zeta^2,Zeta^-2} is a basis? :-s

Also, how does the Galois Group of Q(Zeta) act on this new basis?

Furthermore, how can you prove the existence of a unique subfield E such that [Q(Zeta):E]=2, & find all a,b,c,d such that aZeta+bZeta^-1+cZeta^2+dZeta^-2 is in E but not Q? (apparently you can then show that E = Q(Sqrt(5)) !! :-o )

I have absolutely no idea where to begin! :-s

Any help on ANY of these questions would be amazing! :-)

Thank you! x

Originally Posted by AAM

Hi everyone! :-)

You like to smile a lot, .

Suppose Zeta is any primitive 5th root of unity.

Then we know that Q(Zeta) has a basis {1,Zeta,Zeta^2,Zeta^3} because Zeta is algebraic over Q.

But how can I show that {Zeta,Zeta^-1,Zeta^2,Zeta^-2} is a basis? :-s

Remember that and .

Also, how does the Galois Group of Q(Zeta) act on this new basis?

The Galois group, is isomorphic to the cyclic group . Let be the automorphism defined by . Then . Since you know you can determine what is for .

Furthermore, how can you prove the existence of a unique subfield E such that [Q(Zeta):E]=2, & find all a,b,c,d such that aZeta+bZeta^-1+cZeta^2+dZeta^-2 is in E but not Q? (apparently you can then show that E = Q(Sqrt(5)) !! :-o )

Therefore, is a subfield of .