Abstract: In the research of Hadamard matrices, by using the method of factorization of prime ideals in algebraic number theory, we proved that circulant Hadamard matrices with order 4^r(r＞1) do not exist, and obtained the Hadamard matrices with order 4.For a circulant matrix H satisfying HH^{T=nI with order pr (p is a prime) andintegral elements, we completely determined the structure of H.Such matrices can be viewed as matrices over finite field F_pr, thus we obtained the construction of a sort of orthogonal codes over F_pr.}

Key words: algebraic numbers, circulant matrix, hadamard matrix, cyclotomic fields