One example of inseparable polynomial

\mathbf{Theorem:} F is a field. If F[x] contains an inseparable polynomial, then F can not have characteristic 0 or F is some finite field with characteristic p.

In general, we do not need to worry about the separability of the extension field, because we often deal with finite fields or the one of characteristic 0.

An example of inseparable polynomial
Let K be a field of characteristic p, and F=K(x) is the field of rational polynomials, x is the indeterminate. Consider F[y], y is the indeterminate, x is an irreducible element in this UFD. So f(y)=y^p-x is irreducible in E by the Einsenstein criterion. There exists a splitting field of f(y) over F. Suppose \sigma is root of f(y) then \sigma^p=x. By freshman’s dream f(y)=(y-\sigma)^p. So f has multiple roots. It is not separable.

\mathbf{Remark:} Isaacs, p281

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