On the defining polynomials of maximal real cyclotomic extensions

Defining polynomials of maximal real cyclotomic extensionsSince b i = 1 for i = p ν−1 (p − 2), p ν−1 (p − 4), . . . , p ν−1 , and b i = 0 otherwise, with our precedingnotations, by expanding the determinant according to its last column, we obtain( ) ( )m − j − pa j = (−1) (m−j−pν−1)/2 ν−1m − j − 3pA j+2 + (−1) (m−j−3pν−1)/2 ν−1A j+222( )m − j − kp+ · · · + (−1) (m−j−kpν−1)/2 ν−1A j+2 ,2for j satisfying m − kp ν−1 ≥ j ≥ m − (k + 2)p ν−1 , with k odd.Again from (5), we know that for j = 0, 2,. . . , m, the a j are the solution of the system of m/2 + 1linear equations( ) mb 2m =0Here Cramer’s rule yieldsb 2m−2 =.b m =( mm2a j =∣a m( m1)a m +)a m +( m0)..(mm−2−j2( m m−j2))( m − 20)a m−2( )( m − 20m−2a m−2 + · · · + a 020)∣0 · · · 0 b 2m ∣∣∣∣∣∣∣∣∣∣∣∣∣. .. . .. . .. .. 0 . ,(· · · · · · j+2)0b m+j+2)· · · · · · b m+jwhere b i = 1 for i = p ν−1 (p − 1), p ν−1 (p − 3), . . . , p ν−1 2, 0, and b i = 0 otherwise. It follows thata j = (−1) (m−j)/2 A j+2( m − j2( j+21) ( )m − j − 2p+ (−1) (m−j−2pν−1)/2 ν−1A j+2+ · · · + (−1) (m−j−kpν−1 )/2 A j+2( m − j − kpν−1for m − kp ν−1 ≥ j > m − (k + 2)p ν−1 , with k even.The same reasoning applies to the case p ≡ 3 (mod 4). From (5), it follows that, for j = 1, 3, . . . , m,the a j satisfy the system of linear equations( ) mb 2m = a m0( ) ( )m m − 2b 2m−2 = a m + a m−21 0.b m+1 =( ) mm−1a m +22),( )( m − 21m−3a m−2 + · · · + a 120)1872