Let f Element Fpx be a polynomial and suppose that its irred

Let f Element F_p[x] be a polynomial and suppose that its irreducible factors have degrees {d_1, ..., d_r}. Prove that the splitting field of f has degree lcm(d_1, ..., (d_r}.

Solution

Suffices to prove the statement for r = 2, as the argument goes through for any r.

So let

                                     f(x) = g(x) h(x)

where g and h are irreducible over F=F_p of degrees m and n respectively.

As g is irreducbile over F and is of degree m , its splitting field is F_p^m, the unique extension over F of dimension m.

Similarly, splitting field of h is F_pⁿ, the unique extension over F of dimension n.

Let t = lcm (m,n)

Clearly both g and h , and hence f split in F_p^t .

and t is the smallest such integer , as both m and n have to divide the dimension of the splitting field.

Thus we conclue that the splitting field of f is F_p^t , with t = lcm (m,n)