Algebra: Abstract and Modern

14

Extension Fields

14.1 Extensions of a Field

14.2 Algebraic Extensions

14.3 Algebraically Closed Fields

14.4 Derivatives and Multiple Roots

14.5 Finite 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 ...