For notations and the results we have already proved, see part (1).

Lemma 3.  Let q \geq 1, \ n \geq 2 be integers. Then |\Phi_n(q)| > q-1.

Proof. By definition of \Phi_n, it suffices to prove that |q - \alpha| > q- 1 for any n-th root of unity \alpha \neq 1. So if \alpha = \cos \theta + i \sin \theta, then \cos \theta < 1 and thus

|q - \alpha|^2=q^2-(2\cos \theta)q + 1 > (q-1)^2 \geq q-1. \ \Box

Wedderburn’s Little Theorem. (J. M. Wedderburn, 1905) Every finite division ring is a field.

Proof. Let D be a finite division ring with the center Z. Then Z is a (finite) field and D is a finite dimensional vector space over Z. So if |Z|=q and \dim_Z D=n, then |D|=q^n. If n = 1, then D=Z and we are done. So we will assume that n \geq 2 and we will get a contradiction. Let D^{\times}=D \setminus \{0\}, as usual, be the multiplicative group of D. Clearly Z^{\times}=Z \setminus \{0\} is the center of D^{\times}. Also, for any a \in D, let C(a) be the centralizer of a in D. Then C(a) is also a finite division ring and thus a finite dimensional vector space over Z. Let \dim_Z C(a)=n_a. Then |C(a)|=q^{n_a}. It is clear that the centralizer of a in D^{\times} is C(a)^{\times}=C(a) \setminus \{0\}. So the class equation of D^{\times} gives us

\displaystyle q^n-1=|D^{\times}|=|Z^{\times}| + \sum_{a}[D^{\times}:C(a)^{\times}]=q-1 + \sum_{a} \frac{q^n-1}{q^{n_a}-1}. \ \ \ \ \ \ (\dagger)

By Lemma 1 and Lemma 2 in part (1), |\Phi_n(q)| divides both q^n-1 and \sum_a \displaystyle \frac{q^n-1}{q^{n_a}-1}. So |\Phi_n(q)| \mid q-1, by (\dagger ). Thus |\Phi_n(q)| \leq q-1, contradicting Lemma 3. \Box

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s