I've enjoyed D.E. Littlewood's Skeleton Key to Mathematics up to chapter 8. After chapter 8, increasingly more statements are left without proof and I'm lost.
He writes
[...]it can be shown that the difference algebra $A'$ can be expressed as a direct sum of irreducible algebras, and each algebra is equivalent to the direct product of a matrix algebra and a division algebra. That is to say, each element can be regarded as a matrix of which the elements are not necessarily numbers, but elements of the division algebra. And all such matrices of the elements are the division algebra are elements of the given irreducible algebra. As regards the division algebras, the possible types of these depend upon the field over which the coefficients are taken[...] This explains why quaternions over the field of complex numbers become $2$ rowed matrix algebras
Could someone explain this?