Sentence:  In the case of the polynomial t^2+1 \in \R[t], the situation is quite clear; its <WSD>splitting field</WSD> is \C since t^2+1=(t+i)(t-i) where i=\sqrt{-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.