ABSTRACT
One of the mathematical problems that incited the development of a theory of field extensions was that of finding roots of polynomials. Given a polynomial f ∈ K[x], one tries to find an element α with f(α) = 0. It may well happen that no such element exists in the given field K, but there is still the chance that a root of f exists in some larger field L ⊇ K; for example, the polynomial f ( x ) = x 2 + 1 ∈ ℝ [ x ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136554/b1e97aa6-f63a-4d22-960d-a98dde0e94f9/content/eq3012.tif"/> has no root in ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136554/b1e97aa6-f63a-4d22-960d-a98dde0e94f9/content/eq3013.tif"/> , but has the roots ±i in ℂ ⊇ ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136554/b1e97aa6-f63a-4d22-960d-a98dde0e94f9/content/eq3014.tif"/> . In fact, it turns out that every polynomial f ∈ K[x] (where K is an arbitrary field) possesses a root in some extension of K. With the right point of view, this is not even hard to prove.
