Exercise 18, § 9

(a) Let $K$ be a field and $a, b$ two non-permutable elements of $K$. Show that (1) $a = (b - (a - 1)^{-1}b(a - 1))(a^{-1}ba - (a - 1)^{-1}b(a - 1))^{-1}$ (2) $a = (1 - (a - 1)^{-1}b^{-1}(a - 1)b)(a^{-1}b^{-1}ab - (a - 1)^{-1}b^{-1}(a - 1)b)^{-1}$.

(b) Let $K$ be a non-commutative field and $x$ an element not belonging to the centre of $K$. Show that $K$ is generated by the set of conjugates $axa^{-1}$ of $x$. (Let $K_1$ be the subfield of $K$ generated by the set of conjugates of $x$. Deduce from (a) that, if $K_1 \neq K$ and $a \in K_1$ and $b \notin K_1$, then necessarily $ab = ba$. Deduce a contradiction by considering in $K_1$ two non-permutable elements $a, a'$ and an element $b \notin K_1$ and noting that $ba \notin K_1$.)

(c) Deduce from (b) that if $Z$ is the centre of $K$ and $H$ is a subfield of $K$ such that $Z \subset H \subset K$ and $aHa^{-1} = H$ for all $a \neq 0$ in $K$, then $H = Z$ or $H = K$ (Cartan-Brauer-Hua Theorem).

(d) Deduce from (a) that in a non-commutative field $K$, the set of commutators $aba^{-1}b^{-1}$ of elements $\neq 0$ in $K$ generates the field $K$.