Let be a prime unequal to the characteristic of the field . Show that, if , then is either irreducible in or it has a root in .

Let be a p-th primitive root of unity, . Since has characteristic 0, contains distinct pth root of unity.

Let . is a multiplicative subgroup of .

Suppose is a splitting field of over and is one root of in . All the roots of are , . So . , , . We can verify that this induces a monomorphism

Since is cyclic of order . which is a subgroup of can only be the trivial group or the whole group .

If , . And splits over . Then it must have a root in .

If , then , must be irreducible in , hence also in .

Let be the cyclotomic field of the pth roots of unity over the field of characteristic 0. Show that can be imbedded in a field which has a root tower over such that the integers are primes and . Call such a root tower normalized.

Since is the cyclotomic field over of characteristic 0, is abelian by lemma 1 in page 252.

Then is a finite solvable group. So has a composition series

whose composition factor is cyclic of prime order . By Galois corresponding theorem, we have an increasing chain of subfields

such that and . So is prime. This is a normalized root tower.

Obtain normalized root tower over of the cyclotomic fields of 5th and 7th roots of unity.

Suppose is a primitive 5th roots of unity. and is multiplicative group of , because maps to itself. The automorphism is a generator of .

the corresponding root tower is

It is easy to verify this is a normalized root tower.

For 7th roots of unity, suppose is a primitive 7th root of unity,

is a normalized tower where .

Prove that, if has a solvable Galois group over a field of characteristic 0. Then its splitting field can be imbedded in an extension field which has a normalized tower over .

Suppose splitting field is and , the proof relies on adjoin nth roots of unity to .

Jacobson p256.