Cyclotomic polynomials and primitive roots of unity

[xuying1209]

w_{n} is primitive root of unity of order n, w_{m} is primitive root of unity of order m,
all primitve roots of unity of order n are roots of Cyclotomic polynomials
phi_{n}(x) which is a minimal polynomial of all primitive roots of unity of order n ,
similarly, phi_{m}(y) is a minimal polynomial of all primitive roots of unity of order m ,
then, what is the minimal polynomial of (W_{n},w_{m}), if exists or no?

Thank you very much! what book I can find some subject about primitve roots of unity.

 

[mathwonk]

are you asking for the minimal polynomial of a product of two roots of 1?

or of the field they generate together?

and I presume you are working over the rationals Q, since yiou assume the cyclotomic polynomials are irreducible.i like van der waerden's old modern algebra, for a basic introduction. probably Gauss's disquisitiones is one of the best sources, but most number theoiry and abstract algebra books will say something, like hungerford, dummit - foote, michael artin, jacobson.

 

[xuying1209]

I want to know the minimal polynomial of the field they generate together?

Thank you !

 

[mathwonk]

have you tried an example? for instance suppose z is a primitive complex 6th root of unity and w is a primitive complex 15th root of unity. then together they belong to the group of let's see 30th roots of unity. If you take say z^3, you get hmmmm -1 i suppose. anyway, it looks as if -w is a primitive 30th root of unity. is that right? then -w^5 sems like a primitive 6 th root of unity, and w a primitive 15th root.

so it seems that together they generate the same field as w does alone.try another one, like 18th root and 12th root. what happens?

 

[mathwonk]

by the way there is no such thing as"the minimal polynomial of a field", you need to choose a generating element first, but here you can always choose it to be another primitive root of unity, perhaps of roder the lcm of the two orders you start with?