Definition. Let D be a division algebra with the center k. We denote by [D,D] the additive subgroup of D generated by the set \{ab-ba : \ a,b \in D \}. Also, we denote by D' the subgroup of D^{\times} generated by the set \{aba^{-1}b^{-1}: \ a,b \in D^{\times} \}.

Lemma. Let r, s be integers, a_1 , \ldots , a_r \in D and b_1, \ldots , b_r \in D'. There exists v \in D' such that (b_1a_1b_2a_2 \ldots b_ra_r)^s=(a_1a_2 \ldots a_r)^s v.

Proof. Apply the following repeatedly: if a \in D and b \in D', then ba = ac, where c = b(b^{-1}a^{-1}ba) \in D'. \ \Box

Theorem. Let D be a finite dimensional central division k-algebra of degree n. For every a \in D there exist u \in [D,D] and v \in D' such that \text{Trd}_D(a)=na + u and \text{Nrd}_D(a)=a^nv.

Proof. Let f(x) be the minimal polynomial of a over k and let m = \deg f(x). Let r = \frac{n}{m}. Then, by the theorem in this post, \text{Prd}_D(a,x)=(f(x))^r and by Wedderburn’s factorization theorem there exist non-zero elements c_1, c_2, \ldots , c_m such that

f(x) = (x - c_1ac_1^{-1})(x - c_2ac_2^{-1}) \ldots (x - c_m a c_m^{-1}) .

Let \alpha = \sum_{i=1}^m c_i a c_i^{-1} and \beta = \prod_{i=1}^m c_i a c_i^{-1}. Then

f(x)=x^m - \alpha x^{m-1} + \ldots + (-1)^m \beta

and so

\text{Prd}_D(a,x) = (x^m - \alpha x^{m-1} + \ldots + (-1)^m \beta)^r= x^n - r \alpha x^{n-1} + \ldots + (-1)^n \beta^r.

Therefore \text{Trd}_D(a) = r \alpha and \text{Nrd}_D(a) = \beta^r. Hence

\text{Trd}_D(a)=r \alpha = r \sum_{i=1}^m c_i ac_i^{-1} = na + r\sum_{i=1}^m(c_iac_i^{-1} - ac_i^{-1}c_i).

Let u =r\sum_{i=1}^m (c_iac_i^{-1} - ac_ic_i^{-1}). Then u \in [D,D] and \text{Trd}_D(a)=na+u. Now let b_i=c_iac_i^{-1}a^{-1}. Then the lemma gives us

\text{Nrd}_D(a)=\beta^r = (b_1ab_2a \ldots b_m a)^r = a^n v,

for some v \in D'. \ \Box


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