Lectures on the Algebraic Theory of Fields - Tata Institute of ...

4. Separability 27f (x). Thenφ(y) is irreducible in k[y] andφ(y) has no repeated roots.Letβ 1 ,...,β t be the roots ofφ(y) inΩ. Thenf (x)=(x pe −β 1 )···(x pe −β t ).Thus n=t· p e . The polynomial x Pe −β i has inΩall roots identicalto one of them sayα i . Thenx Pe −β i = x Pe −∆α pei= (x−α i ) peThusf (x)={(x−α 1 )···(x−α t )} peMoreover sinceβ 1 ...β t are distinct, α 1 ,...,α t are also distinct.Hence9) Over a field of characteristic po, the roots of an irreducible polynomialare repeated equally often, the multiplicity of a root being p e ,e≥o.It is important to note that (x−α 1 )...(x−α t ) is not a polynomial in 32k[x] and t is not necessarily prime to p.We call t the reduced degree of f (x) (or of any of its roots ) and p e ,its degree of inseparability. ThusDegree of: Reduced degree X-degree of inseparabilityIfω∈Ω then we had seen earlier that k(ω)/k has as many distinctisomorphisms inΩas there are distinct roots inΩof the minimumpolynomial ofωover k. If we call the reduced degree of k(ω) as thekreduced degree ofω we have10) Reduced degree ofω = Number of distinct roots of the minimumpolynomial ofωover k.We may now call a polynomial separable if and only if every root ofit inΩis separable. In particular if f (x)∈k[x] is irreducible thenf (x) is separable if one root of it is separable.