Let , where is a squarefree integer. Let be in , and consider the basis of over . Compute the matrix of the -linear transformation ‘multiplication by ‘ (described previously) with respect to . Give an explicit embedding of in the ring .
We have and . Making these the columns of a matrix , we have , and this is the matrix of with respect to . As we showed in the exercise linked above, is an embedding of in .
Compare to this previous exercise about .