## Tag Archives: separable

### 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