Exercise 9, § 2

(a) Show that the algebra $E$ of the dihedral group $D_4$ of order 8 (I, § 6, Exercise 4) over a field $K$ in which $2 \neq 0$ is isomorphic to the product of four fields isomorphic to $K$ and the algebra of matrices of order 2 over $K$. (If $a, b$ are the two generators of $D_4$ introduced in I, § 6, Exercise 4, the elements of $D_4$ can be written as $a^i b^j$ with $0 \leq i \leq 3, 0 \leq j \leq 1$; consider the basis of $E$ consisting of the four elements $\frac{1}{2}(e + a^2), \frac{1}{2}(e - a^2), \frac{1}{2}(a + a^3), \frac{1}{2}(a - a^3)$ and these four elements multiplied on the right by $b$; use Exercise 4.) Deduce that, if $-1$ is a square in $K$, the algebras over $K$ of the non-isomorphic groups $Q$ and $D_4$ are isomorphic.

(b) Show similarly that the algebra $F$ of the dihedral group $D_3$ of order 6 over a field $K$ in which $2 \neq 0$ and $3 \neq 0$ is isomorphic to the product of two fields isomorphic to $K$ and the matrix algebra of order 2 over $K$ (consider here the basis of F consisting of the elements $e + a + a^2, a + a^2 - 2e, a - a^2$ and these three elements multiplied on the right by $b$).