Sentence:  Example 4.3. By definition, \Q(i) is the <WSD>splitting field</WSD> of x^2+1 over \Q, and \C is the <WSD>splitting field</WSD> for the same polynomial, over \R. / Example 4.4. The <WSD>splitting field</WSD> F of x^8-1 over \Q is generated by \zeta:=2^{2π i/8}; indeed, the roots of x^8-1 are all the 8-th roots of 1, and all of them are powers of \zeta:...In fact, \zeta is a root of the polynomial x^4+1, which is irreducible over \Q; therefore F=\Q(\zeta) is 'already' the <WSD>splitting field</WSD> of x^4+1. 
Meaning of the word marked with <WSD> tags:  Given a polynomial "p" over a field "K", the smallest extension field "L" of "K" such that "p", as a polynomial over "L", decomposes into linear factors (polynomials of degree 1);  given a set "P" of polynomials over "K", the smallest extension field of "K" over which every polynomial in "P" decomposes into linear factors.