Cyclotomic polynomials and primitive roots of unity

In summary, the conversation discusses primitive roots of unity and their minimal polynomials. It is mentioned that all primitive roots of unity of a certain order are roots of cyclotomic polynomials and that the minimal polynomials of such roots are the same. The conversation also mentions the minimal polynomial of a product of two primitive roots of unity and the field they generate together. Some recommended sources for studying primitive roots of unity are also mentioned.
  • #1
xuying1209
4
0
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.:wink:
 
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  • #2
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.
 
  • #3
I want to know the minimal polynomial of the field they generate together?

Thank you !:wink:
 
  • #4
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?
 
  • #5
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?
 

1. What are cyclotomic polynomials?

Cyclotomic polynomials are a type of polynomial that are defined by their roots, which are the primitive roots of unity. They have coefficients that are either 0 or 1, and their degree is equal to the totient function of the order of the root of unity.

2. What is a primitive root of unity?

A primitive root of unity is a complex number that, when raised to certain powers, will produce all the other roots of unity. These roots are important in number theory and have various applications in mathematics, especially in the study of cyclotomic polynomials.

3. What is the relationship between cyclotomic polynomials and primitive roots of unity?

Cyclotomic polynomials are defined by their roots, which are the primitive roots of unity. In other words, the roots of a cyclotomic polynomial are the numbers that, when raised to certain powers, will produce all the other roots of unity. Thus, the two concepts are closely related and often used together in mathematics.

4. How are cyclotomic polynomials used in number theory?

Cyclotomic polynomials have various applications in number theory, such as in the study of prime numbers, factorization, and the distribution of primes. They are also used in algebraic number theory and algebraic geometry.

5. Can cyclotomic polynomials be computed efficiently?

Yes, there are algorithms and formulas that can be used to compute cyclotomic polynomials efficiently. One such method is the recursive formula known as the Möbius inversion formula. Additionally, there are also specific algorithms for computing cyclotomic polynomials of certain orders, such as the Fast Fourier Transform (FFT) algorithm for polynomials of order 2^n.

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