Thread: Galois group

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 ?

Originally Posted by MARMAR

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 ?

I'll get right on that.

I think you mean "S3 or C3"? What have you tried so far? What does irreducibility tell you? What about separability? What is the significance of the Galois group being S3 versus C3?

Originally Posted by MARMAR

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

Originally Posted by TheChaz

I'll get right on that.

yes. sorry i mean S3 or C3

Originally Posted by tonio

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

i need more clarification pls

If what tonio wrote is not clear to you, then you need to re-read the relevant section of your textbook. What he wrote is very basic.

Originally Posted by MARMAR

i need more clarification pls

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

Originally Posted by tonio

Do a little self work on case (1): How does the extension look(Galois extension) ? What's its degree 3? 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?-(-i,i)

If you get stuck write back, and if you don't then you can see how to work on case (2).

Tonio