Exercise 7, § 2

Let $K$ be a commutative field in which $2 \neq 0$ and $E$ the algebra over $K$ with a basis of four elements $1, i, j, k$ (1 the unit element) with multiplication table

Show that $E$ is isomorphic to the product of four fields isomorphic to $K$ (consider the basis of $E$ consisting of the elements $(1 + \varepsilon i)(1 + \varepsilon' j)$, where $\varepsilon$ and $\varepsilon'$ are equal to 1 or $-1$).

The algebra $E$ is the algebra (over $K$) of the product of two cyclic groups of order 2. Generalize this to the algebra of the product group of $n$ cyclic groups of order 2 (cf. § 4, no. 1, Example 2).