Abstract Algebra Theory and Applications - Computer Science ...

20.2 POLYNOMIAL CODES 357Expanding this determinant by cofactors on the last column, we see thatp(x) is a polynomial of at most degree n−1. Moreover, the roots of p(x) areα 1 , . . . , α n−1 , since the substitution of any one of these elements in the lastcolumn will produce a column identical to the last column in the matrix.Remember that the determinant of a matrix is zero if it has two identicalcolumns. Therefore,wherep(x) = (x − α 1 )(x − α 2 ) · · · (x − α n−1 )β,⎛β = (−1) n+n det⎜⎝By our induction hypothesis,β = (−1) n+n⎞1 1 · · · 1α 1 α 2 · · · α n−1α1 2 α2 2 · · · αn−12 ... . ..⎟. ⎠α1 n−2 α2 n−2 · · · αn−1n−2∏1≤j