四元二次多项式可约的充要条件

Necessary and Sufficient Conditions of Reducibility for Quadratic Polynomial with Four Elements

应用正交变换研究实系数四元二次多项式于实数域及复数域可约的判定方法,通过正交变换将一般的实系数四元二次多项式于实数域及复数域的可约性等价转化为只含有平方项、一次项与常数项的实系数二次多项式的可约性,将实系数四元二次多项式具体分解为齐二次一元多项式、齐二次二元多项式、齐二次三元多项式、齐二次四元多项式、非齐次一元二次多项式、非齐次二元二次多项式、非齐次三元二次多项式、非齐次四元二次多项式,构造由这八个多项式的系数组成的行列式满足的关系式来刻画实系数四元二次多项式于实数域及复数域可约的充要条件,应用对称矩阵的合同变换给出实系数四元二次多项式的因式分解,拓宽了已有文献的研究结果。

Determinate methods of reducibility for quadratic polynomial of four elements with real coefficients in real or complex field are studied by using orthogonal transform.Reducibility for quadratic polynomial of four elements with real coefficients in real or complex field is equivalence to reducibility for quadratic polynomial of four elements with quadratic term,linear term and constant term.Quadratic polynomial of four elements is decomposed homogeneous quadratic polynomial of one element,homogeneous quadratic polynomial of two elements,homogeneous quadratic polynomial of three elements,homogeneous quadratic polynomial of four elements,non-homogeneous quadratic polynomial of one element,non-homogeneous quadratic polynomial of two elements,non-homogeneous quadratic polynomial of three elements,non-homogeneous quadratic polynomial of four elements.Some necessary and sufficient conditions of reducibility for quadratic polynomial with four elements are obtained in real or complex field by using orthogonal transforms of the real coefficients of eight polynomials.Factorization of quadratic polynomial of four elements with real coefficients is given by using congruent transformation of symmetric matrix.These results generalize some known results.