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Algebra: Abstract and Modern by U. M. Swamy, A. V. S. N. Murthy

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14

Extension Fields

It is well known that the field ℚ of rational numbers is a subfield ℝ of the field of real numbers and, in this case, we say that ℝ is an extension field of ℚ. Likewise, the field ℂ of complex numbers is an extension of ℝ. Let us recall that the polynomial 1 + x2 has no root in ℝ. However, there is an extension field, namely ℂ, containing a root of 1 + x2. In this chapter, we discuss in detail about the existence of an extension field containing roots of a given polynomial over a given field.

The field ℝ of real numbers has a deficit that not all polynomials over ℝ have roots in ℝ. The ...

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