**Definition**. Let be a division algebra with the center We denote by the additive subgroup of generated by the set Also, we denote by the subgroup of generated by the set

**Lemma**. Let be integers, and There exists such that

*Proof*. Apply the following repeatedly: if and then where

**Theorem**. Let be a finite dimensional central division -algebra of degree For every there exist and such that and

*Proof*. Let be the minimal polynomial of over and let Let Then, by the theorem in this post, and by Wedderburn’s factorization theorem there exist non-zero elements such that

Let and Then

and so

Therefore and Hence

Let Then and Now let Then the lemma gives us

for some

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