Equations in Quaternions
Equations in Quaternions
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DOI：
10.1080/00029890.1941.11991158
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发表时间：
1941-12
影响因子：
0.5
通讯作者：
I. Niven
中科院分区：
文献类型：
--
作者：
I. Niven
(1)系数来自于真实的四元数的代数。作者在m为奇数时证明了这一结果，但当Nathan Jacobson指出该结果（不限制m）可以作为Ore [1]的某些工作的简单结果而得到时，该证明被废弃。这一点在§ 2中有详细说明。在§ 3中，我们给出了一种求（1）的根的方法，它不太实用，因为它涉及到同时求解两个2 m-1次的真实的方程。所用的方法是对应于（1）的二次方程的西尔维斯特处理[2]的推广。西尔维斯特关于一个二次方程有六个根的结论是不正确的，因为他忽略了证明它们的存在，也忽略了无穷多个根的可能性;一个完整的分析在§ 4（定理2）中给出。（1）的根的个数在§ 5中讨论（定理3），给出了无穷多个根的充要条件，本文给出的（1）的根存在的证明是在方程系数为实闭域R上的四元数的一般情况下给出的（即R中没有平方和等于-1，R的代数扩张也没有这个性质）。
(1) with coefficients from the algebra of real quaternions. The writer had proved this result when m is odd, but the proof was rendered obsolete when Nathan Jacobson pointed out that the result (without restriction on m) can be obtained as a simple consequence of some work of Ore [1]. This is given in detail in § 2. In § 3 we give a method for obtaining the roots of (1), which is not very practical in the sense that it involves the simultaneous solving of two real equations of degree 2m-1. The method used is a generalization of Sylvester's treatment [2] of the quadratic equation corresponding to (1). Sylvester's conclusion that a quadratic equation has six roots is incorrect because he neglects to show that they exist, and also overlooks the possibility of an infinite number of roots; a complete analysis is~ iven in § 4 (Theorem 2). The number of roots of (1) is discussed in § 5 (Theorem 3), necessary and sufficient conditions being given for an infinite number of roots.The proof given here of the existence of a root of (1) is stated for the general case where the coefficients of the equation are quaternions over any real-closed field R (ie, no sum of squares in R is equal to-1, and no algebraic extension of R has this property).