**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*

*(ii)** K* = *F*(*α*_{1},…,*α*_{n})*; that is, K is generated over F by the roots α*_{1},…,*α*_{n }*of ƒ*(*x*)* in K*.

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

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