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.