Galois theory with non-irreducible polynomials I was browsing through Wikipedia for some reason or another, and it says that there is a way to do Galois theory without worrying about whether or not polynomials are irreducible.  That's a bit surprising to me.
Is Wikipedia wrong (as is often the case when I read something surprising)?  If not, where can I hear more about it?  
 A: There is a very big difference between "you can do Galois theory without worrying about whether or not a polynomial is irreducible" and "one can do Galois theory while working with polynomials that are not irreducible". The former implies that the results will hold regardless of whether the polynomial is irreducible or not; the latter merely says that one can adjust the work so as to deal with reducible polynomials.
If $f(x)$ is not irreducible, but factors as $f(x)=p_1(x)^{a_1}\cdots p_k(x)^{a_k}$ with $p_j(x)$ irreducible and separable, then you can certainly still talk about the Galois group of the splitting field of $f(x)$, and it will have all the usual properties: it would simply be compositum of the splitting fields of the different $p_i$, and so the Galois group of $f$ would have normal subgroups corresponding to the splitting fields of each $p_i$. 
When the polynomials are inseparable one runs into greater difficulties; the usual technique is to separate the extension into a separable extension and a purely inseparable extension. For the separable ones, the usual techniques of Galois Theory work; for the purely inseparable ones, we have special results that apply.
That said, the quote from Wikipedia is rather poor, since separability is a property of the irreducible factors of the polynomials. Repeated factors do not affect separability or lack thereof. But even if we were to take it at face value, it does not claim that one can do Galois theory "without worrying"; it would say that if you are doing Galois theory with non-irreducible polynomials (in the way that one usually does: by splitting the polynomial into irreducible factors), then you would not worry about whether the polynomial itself has repeated roots or not, because repeated roots could come from repeated separable irreducible factors. 
