2005/09/11 | 代数元素[Algebraic Element]
类别(∑〖数学〗) | 评论(0) | 阅读(34) | 发表于 10:39
Given a field and an extension field , an element is called algebraic over if it is a root of some nonzero polynomial with coefficients in .

Obviously, every element of is algebraic over . Moreover, the sum, difference, product, and quotient of algebraic elements are again algebraic. It follows that the simple extension field is an algebraic extension of iff is algebraic over .

The imaginary unit i is algebraic over the field of real numbers since it is a root of the polynomial . Because its coefficients are integers, it is even true that is algebraic over the field of rational numbers, i.e., it is an algebraic number (and also an algebraic integer). As a consequence, and are algebraic extensions of and respectively. (Here, is the complex field , whereas is the total ring of fractions of the ring of Gaussian integers .)
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