Let K be a field extension and let f \in K{X} be irreducible and separable of degree 3. show that the Galois group of f is isomorphic to S3 to C3 (the cyclic group of order
3)
*Determine the Galois group of X^3-X-1 over Q ?
What does "isomorphic to S_3 to C_3" mean?? Did you mean "or" instead of "to" there?
If so, you may want to compare the situations (1) the extension field K[x]/<f(x)> only contains one root of f(x),
and (2) The field K[x]/<f(x)> contains all the three different roots of f(x).
Tonio
Do a little self work on case (1): How does the extension look? What's its degree? Take into account that if an extension contains two of the pol's
roots then it MUST contain the third one as well (why? Let w be one of your irreducible pol's roots, and write f(x) = (x-w)q(x) , with q(x) a
quadratic pol. What're this quadratic's roots?.
If you get stuck write back, and if you don't then you can see how to work on case (2).
Tonio