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Basic Abstract Algebra by S. K. Jain, S. R. Nagpaul, P. B. Bhattacharya

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CHAPTER 16

Normal and separable extensions

1 Splitting fields

Definition. Let ƒ(x) be a polynomial in F[x] of degree ≥ 1. Then an extension K of F is called a splitting field of ƒ(x) over F if

(i) ƒ(x) factors into linear factors in K[x]; that is

images

(ii) K = F(α1,…,αn); that is, K is generated over F by the roots α1,…,αn of ƒ(x) in K.

For example, (i) the field images is a splitting field of images over Q; (ii) a splitting field of over R is the field ...

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