Can Cosine or Sine Values be Expressed Using Roots of Rationals?

[gonzo]

This is indirectly related to issues with cyclotomic polynomials and glaois groups.

Is there some easy way to know if you are dealing with a cos or sin that is expressable in terms of roots of rationals? Like $\pi/3$ for example? If so, is there any straightforward way of figuring it out?

I'm asking because I'm trying to do a problem involving the galois group for the 7th roots of unity. The fact that there is a order 2 and order 3 subgroup of the galois group makes me think that Q(i) must be a fixed field and then some cube root another one for the subgroups. This would further imply that $cos(4\pi/7)$ is expressable as a ratioal function plus a cube root (unless I'm way off in space, which is possible).

Thanks

 

[leach]

A very interesting problem. I'm not very fluent with Galois theory these days, so please take my words with a grain of salt.

For the 7th roots of unity we have the polynomial

$$(x-1)^7 \quad = \quad (x-1)(x^6+x^5+x^4+x^3+x^2+x+1)$$

So the interesting polynomial for these roots is:

$$p \quad = \quad x^6+x^5+x^4+x^3+x^2+x+1$$

Now, it is obvious that this polynomial has as Galois group $$\mathbb{Z}_7$$. If we call $$\omega$$ some seventh root of unity (not 1), the other roots of $$p$$ are the powers $$\omega^k,\, k=2,\ldots, 6$$. Hence the field for this polynomial is $$\mathbb{Q}(\omega)$$, extension of dimension 7 over $$\mathbb{Q}$$, and its automorphisms are obviously the powers of the roots.

The group $$\mathbb{Z}_7$$ has no nontrivial subgroups. I think you are erroneously taking $$\mathbb{Z}_6$$ as the Galois group of the polynomial p.

Anyway, we come to the conclusion that any root of our polynomial p has the form $$a_0 + a_1\omega + a_2\omega^2 + a_3\omega^3 + a_4\omega^4 + a_5\omega^5 + a_6\omega^6$$, where the $$a_i$$ are rationals. Since $$\omega$$ is a radical (a seventh root of unity), this trivially means that all the roots of p are radicals.

However, I think that in order to get better information on the structure of the roots, the coefficients $$a_i$$, etc., you could try reading modern elaborations of Gauss' works on cyclotomic groups. I hope this link helps : http://en.wikipedia.org/wiki/Gaussian_period

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If you take sixth roots of unity, you have $$\mathbb{Z}_6 \cong \mathbb{Z}_3\times\mathbb{Z}_2$$, so the Galois group should be generated by two radicals $$\mathbb{Q}(i,\xi)$$, with $$\xi^3 = 1$$.

This means that the sixth roots of unity should be writeable in the form: $$a_0 + a_1 i + a_2 \xi + a_3 i\xi + a_4\xi^2 + a_5i\xi^2$$. Again, more finese in the form of these roots should be achieved turning to Gauss' works.

 

[tacman]

for your question! Yes, there is a way to determine if a cosine or sine can be expressed in terms of roots of rationals. This is directly related to cyclotomic polynomials and their roots, also known as primitive roots of unity.

First, let's define a primitive root of unity as a complex number that, when raised to a power, equals 1. For example, the complex number e^(2πi/3) is a primitive third root of unity because (e^(2πi/3))^3 = 1.

Now, let's look at the roots of the cyclotomic polynomial Φ_n(x), where n is any positive integer. These roots are exactly all the primitive nth roots of unity. For example, the roots of Φ_3(x) are e^(2πi/3) and e^(4πi/3), which are the primitive third roots of unity.

With this in mind, we can see that if a cosine or sine can be expressed in terms of roots of rationals, it must also be expressible in terms of primitive roots of unity. This is because any rational number can be written as a ratio of two integers, and any integer can be written as a product of powers of primes. Therefore, any rational number can be written as a product of powers of primes, which can then be written as a product of primitive roots of unity.

In your example of cos(4π/7), we can see that this can be expressed in terms of a primitive seventh root of unity, e^(2πi/7). This is because cos(4π/7) = Re(e^(4πi/7)) = Re((e^(2πi/7))^2) = Re((e^(2πi/7))^2), which is a rational function of e^(2πi/7).

In general, if a cosine or sine can be expressed in terms of roots of rationals, it must also be expressible in terms of primitive roots of unity, and therefore can be written as a rational function of these roots. This is directly related to the Galois group of the roots of unity, as you mentioned. The Galois group of the nth roots of unity is isomorphic to the multiplicative group of all nth roots of unity, which means that the Galois group is generated by the primitive nth roots of unity.

In summary