Subfields of Cyclotomic Fields: Examples (Algebra 3: Lecture 25 Video 3)

Lecture 25: We started this lecture by reviewing some things we already know about cyclotomic fields.  We then saw that Gal(Q(zeta_n)/Q) is isomorphic to (Z/nZ)*. We discussed an extended example with n =5.  We saw how to use the Galois correspondence to find the unique nontrivial subfield of Q(zeta_5) containing Q.  We determined when the primitive nth roots of unity form a basis for the nth cyclotomic field.  In order to do this we proved a result about bases for the composite of two Galois extensions, and we discussed composites and intersections of cyclotomic fields.  We then saw how to use the Galois correspondence to determine the subfields of Q(zeta_p). We solved two Algebra Qualifying Exam problems focused on the particular case of Q(zeta_7).  At the end of the lecture we saw how to understand Gal(Q(zeta_n)/Q) in terms of the prime factorization of n.  We mentioned some further topics including the Inverse Galois Problem and the Kronecker-Weber Theorem.

Reading: In this lecture we covered the material of Section 14.5 with some changes.  We skipped over the example involving Q(zeta_13), instead solving two Algebra Qualifying Exam problems about Q(zeta_7) (Fall 2016 #6 and Spring 2013 #5b).  We only briefly mentioned the application given as Corollary 28 and we did not discuss the application to constructibility of the regular n-gon.
We mentioned several exercises that you may want to read (or maybe even try and solve): Exercise 7 in Section 14.7, Exercise 33 in Section 14.6, and Exercise 13 in Section 14.5.