Since

is not perfect it means

and so there exists

so that

. Define the polynomial

. Let

be an extension field having a zero

of

. Notice that if

is a zero of this polynomial then

, but this means

therefore we have shown that

splits over any extension field that has a zero of

. This means either

is irreducible or it factors into linear factors (this is a result about polynomials of prime degree, if you do not know I can prove it for you) over

. It cannot factor into linear factors since it has no zeros in

and so

is irreducible. Thus,

is an inseperable extension. In fact, what we have constructed is a

purely inseperable extension, this is like the worst case scenario of seperability.