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在《数学通报》1964年第1期上,作者撰写了“关于二次最简根式的问題”一文,并且指出了二次最简根式,…,的有理系数多项式f进一步简化的可能性。但此处所谓简化,是指简化成几个二次最简根式的另一有理系数多项式g,而g比f含有较少的项。超过这个范围,即非所论及。我们自然可以提出一个更一般的问題:设f是最简根式,…,的有理系数多项式,问是否存在最简根式,…,的有理系数多项式g,g等于f但比f含有更少的项?本文只讨论属于这个问题的一个方面,就正于读者。在以下,所谓线性相关、线性无关、线性组合等等,如无特别声明,乃指以有理数为系数;并且以R代表有理数体。为以后讨论之用,先将高等代数中一个关于代数扩张的重要定理叙述如下(証明从略):
In “The Mathematics Bulletin,” No. 1, 1964, the author wrote “On the second most simple root problem” article, and pointed out that the second most simple root formula, ..., the rational coefficient polynomial f further simplify the possibility. However, the so-called simplification herein refers to another rational coefficient polynomial g that is reduced to several quadratic simplest roots, and g contains fewer items than f. Exceeding this range is not covered. We can naturally ask a more general question: Let f be the simplest root polynomial of rational coefficients,..., Ask if there is a simplest root formula,..., The rational polynomial of g, g is equal to f but contains fewer terms than f This article only discusses one aspect that belongs to this issue and is just for the reader. In the following, linear correlation, linear independence, linear combination, etc., unless otherwise specified, refer to a rational number as a coefficient; and R represents a rational number. For the purpose of future discussion, we first describe an important theorem about algebraic expansion in higher algebra as follows (the proof is abbreviated):