I was just looking for some clarification regarding the definition of the Galois group of a polynomial $f(x)$. So, if I remember correctly, this is defined as the Galois group of a splitting field of $f(x)$. However, many books I've checked only give the definition for a separable polynomial (i.e. Dummit and Foote).

So, let's say the polynomial is not separable, and let us assume the coefficients are over a perfect field. Then, if we factor $f(x)=p_1^{n_1}(x) ...p_m^{n_m}(x)$, where the factors are irreducible and distinct, then we can write $g(x)=p_1(x) ...p_m(x)$. Since the factors are irreducible and the field is perfect, distinct irreducibles have distinct roots, hence the splitting fields for $f(x)$ and $g(x)$ are the same, hence we can compute the Galois group of $g(x)$. Is this the correct procedure for solving a problem like this?

  • $\begingroup$ If I recall correctly Dummite and Foote define that a polynomial is separable iff every irreducible factor of it is separable so $f$ is separable iff $g$ is by this definition $\endgroup$ – Belgi Aug 15 '14 at 17:58
  • $\begingroup$ The definition they give is just a polynomial which does not have any multiple roots. $\endgroup$ – TheManWhoNeverSleeps Aug 15 '14 at 18:04

Yes, your procedure is correct. Note that you have shown that there is no need to assume that $f(x)$ is separable in the definition of the Galois group.

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