I need to prove if $\mathbb{Q}(\sqrt{3},i)$ is a Galois extension or not. I thought that $\mathbb{Q}(\sqrt{3},i)$ is the splitting field of $(x^2-3)(x^{2}+1)$ so it is separable, but I am not sure about it being normal; I think that it is. Could someone give some help?

  • 2
    $\begingroup$ Hint: Look at the definition of a normal extension again. Every finite extension of $\Bbb{Q}$ is separable. $\endgroup$ – user38268 Feb 19 '13 at 2:11
  • $\begingroup$ Dear @Dimitri, I'm guessing your definition of Galois is separable and normal. As Gerry Myerson indicates in his answer, splitting fields are normal, but this is a non-trivial fact. The converse, however, that normal extensions are splitting fields, is immediate from the definitions. $\endgroup$ – Keenan Kidwell Feb 19 '13 at 2:14
  • $\begingroup$ Thanks, yes i know that splitting fields are normal, maybe i got confuse because i started with other descrition of the extension and then i see the fact that was the splitting field of a polynomial. Thanks for the answer to all. $\endgroup$ – Dimitri Feb 19 '13 at 2:22

If it's a splitting field, it's guaranteed to be normal. Since we are in characteristic zero, it's guaranteed to be separable.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.