Field Extensions and Kronecker's Theorem (Fundamental Theorem of Field Theory), including Examples

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We start with the definition of an extension field E of a (base) field F (note that F is a subfield of E). As an example, The field ℝ[x]/＜x^2+1＞ is a field extension of ℝ that contains a root (zero) of x^2+1. It is also isomorphic to the field of complex numbers ℂ. Why? These ideas can be generalized with the Fundamental Theorem of Field Theory (Kronecker's Theorem). Next, we consider a 5th degree polynomial f(x) in ℤ3[x] that is reducible as the product of two irreducible factors, one of which is a quadratic and one of which is a cubic. There are two field extensions of ℤ3 which contains a root of f(x). One extension field has 3^2 = 9 elements and the other has 3^3 = 27 elements. Coset calculations are key to all of this!

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(0:00) Extension fields gave me difficulty on my grad school Ph.D. prelims
(0:58) Extension field definition
(3:28) Field ℝ[x]/＜x^2+1＞ contains a zero of x^2+1
(13:18) ℝ[x]/＜x^2+1＞ is field isomorphic to ℂ (field of complex numbers)
(15:37) Kronecker's Theorem
(16:30) Example: 5th degree polynomial f(x) in ℤ3[x]

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